' rr. j ELEMENTS OF " \ . ■ ■ vi* * r r\*-- f-r • g VECTOR ANALYSIS : S§H ■ .tKA* Arranged for the use of Students in Physics -a By J. WILLARD GIBBS, Professor of Mathematical Physics in Yale College. ;r S NOT PUBLISHED. NEW HAVEN: Printed by Tuttle, Morehouse & Taylor 1881-4. 1 ' / s'V - . tv: l 1 ' |. , -- ■ •- ■ •• v ■ .>• ...'V , •• "ttt v-tt Umm ■ ■■ ■■ ■ ‘ ’ v r : '/> * ,■% • . ,t V*W 1 11 ■ j ' ,/v fc. % : ; :«r s ' sM v.;m, t? ' f OH t:jgf '; ‘ st ' IS® ELEMENTS OF VECTOR ANALYSIS. By J. Willard Gibbs. [The fundamental principles of the following analysis are such as are familiar under a slightly different form to students of quaternions. The manner in which the subject is developed is somewhat different from that followed in treatises on. quaternions, since the object of the writer does not require any use-of the con¬ ception of the quaternion, being simply to give a suitable notation for those rela¬ tions between vectors, or between vectors and scalars, which seem most import¬ ant, and which lend themselves most readily to analytical transformations, and to explain some of theSe transformations. As a precedent for such a departure from quaternionic usage, Clifford’s Kinematic may be cited. In this connection, the name of Grassmann may also be mentioned, to whose system the following method attaches itself in some respects more closely than to that of Hamilton.] CHAPTER I. COHCERNIHG THE ALGEBRA OF VECTORS. Fundamental Notions. 1. Definition .—If anything lias magnitude and direction, its magnitude and direction taken together constitute what is called a vector. The numerical description of a vector requires three num¬ bers, hut nothing prevents us from using a single letter for its symbolical designation. An algebra or analytical method in which a single letter or other expression is used to specify- a vector may be called a vector algebra or vector analysis. - Def . — As distinguished from vectors the real (positive or negative) quantities of ordinary algebra are called scalars .* As it is convenient that the form of the letter should indicate whether a vector or a scalar is denoted, we shall use the small * The imaginaries of ordinary algebra may be called biscalars, and that which corresponds to them in the theory of vectors, bivectors. But we shall have no occasion to consider either of these. 2 VECTOR ANALYSIS. Greek letters to denote vectors, and the small English letters to denote scalars. (The three letters, i, j, k, will make an excep¬ tion, to he jnentioned more particularly hereafter. Moreover, 7 r will he used in its usual scalar sense, to denote the ratio of the circumference of a circle to its diameter.) 2. Def. —-Vectors are said to he equal when they are the same both in direction and in magnitude. This equality is denoted by the ordinary sign, as a—f The reader will ob¬ serve that this vector equation is the equivalent of three scalar equations. A vector is said to be equal to zero, when its magnitude is zero. Such vectors may be set equal to one another, irrespec¬ tively of any considerations relating to direction. 3. Perhaps the most simple example of a vector is afforded by a directed straight line, as the line drawn from A to B. We may use the notation AB to denote this line as a vector, i. e., to denote its length and direction without regard to its position in other respects. The points A and B may be dis¬ tinguished as the origin and the terminus of the vector. Since any magnitude may be represented by a length, any vector may be represented by a directed line; and it will often be convenient to use language relating to vectors, which refers to them as thus represented. Reversal of Direction , Scalar Multiplication and Division. 4. The negative sign (—) reverses the direction of a vector. (Sometimes the sign -f may be used to call attention to the fact that the vector has not the negative sign.) Def. —A vector is said to be multiplied or divided by a scalar when its magnitude is multiplied or divided by the numerical value of the scalar and its direction is either un¬ changed or reversed according as the scalar is positive or nega¬ tive. These operations are represented by the same methods as multiplication and division in algebra, and are to be regarded as substantially identical with them. The terms scalar multi- plication and scalar division are used to denote multiplication and division by scalars, whether the quantity multiplied or divided is a scalar or a vector. 5. Def. —A unit vector is a vector of which the magnitude is unity. Any vector may be regarded as the product of a positive scalar (the magnitude of the vector) and a unit vector. . The notation « 0 may be used to denote the magnitude of the vector a. VECTOR ANALYSIS. 3 Addition and Subtraction of Vectors. 6. Def.—r The sum of the vectors a, /?, Ac. (written a+p+ Ac.) is the vector found by the following process. Assuming any point A, we determine successively the points B, C, Ac., so that AB=a, BO=/9, Ac. The vector drawn from A to the last point thus determined is the sum required. This is some¬ times called the geometrical sum, to distinguish it from an algebraic sum or an arithmetical sum. It is also called the resultant, and «, /?, Ac., are called the components. When the vectors to be added are all parallel to the same straight line, geometrical addition reduces to algebraic: when they have all the same direction, geometrical addition like algebraic reduces to arithmetical. It may easily be shown that the value of a sum is not affected by changing the order of two consecutive terms, and therefore that it is not affected by any change in the order of the terms. Again, it is evident from the definition that the value of a sum is not altered by uniting any of its terms in brackets, as a+[ft+f\+ Ac., which is in effect to substi¬ tute the sum of the terms enclosed for the terms themselves among the vectors to be added. In other words, the commu¬ tative and associative principles of arithmetical and algebraic addition hold true of geometrical addition. 7. Def. — A vector is said to be subtracted when it is added after reversal of direction. This is indicated by the use of the sign — instead of +. 8 . It is easily shown that the distributive principle of arith¬ metical and algebraic multiplication applies to the multiplica¬ tion of sums of vectors by scalars or sums of scalars : — i. e., (m + w + Ac.) [ar + /3+&c.]=ma + wa-(-&c. + mfi + n/3 + &c. + Ac. 9. Vector Equations. —If we have equations between sums and differences of vectors, we may transpose terms in them, multiply or divide by any scalar, and add or subtract the equa¬ tions, precisely as in the case of the equations of ordinary algebra. Hence, if we have several such equations containing known and unknown vectors, the processes of elimination and reduction by which the unknown vectors may be expressed in terms of the known are precisely the same, and subject to the same limitations, as if the letters representing vectors repre¬ sented scalars. This will be evident if we consider that in the multiplications incident to elimination in the supposed scalar equations the multipliers are the coefficients of the unknown quantities, or functions of these coefficients, and that such 4 VECTOR ANALYSIS. multiplications may be applied to the vector equations, since the coefficients are scalars. 10. Linear relation of four vectors , Coordinates. — If a, /?, and y are any given vectors not parallel to the same plane, any other vector p may he expressed in the form p=aa + b/3+cy. If a, /?, and y are unit vectors, a, b, and c are the ordinary scalar components of p parallel to «, /9, and y. If y—OP, («, f y being unit vectors,) a, b, and c are the cartesian coordi¬ nates of the point P referred to axes through O parallel to a , /9, and y. When the values of these scalars are given, p is said to be given in terms of «, /2, and y. It is generally in this way that the value of a vector is specified, viz., in terms of three known vectors. For such purposes of reference, a sys¬ tem of three mutually perpendicular vectors have certain evi¬ dent advantages. 11 . Normal systems of unit vectors. — The letters i. j, k are appropriated to the designation of a normal system of unit vectors , i. e., three unit vectors, each of'which is at right angles to the other two and determined in direction by them in a perfectly definite manner. We shall always suppose that k is on the side of the i-j plane on which a rotation from i to j (through one right angle) appears counter-clock-wise. In other words, the directions of i, j, and k are to be so determined that if they be turned (remaining rigidly connected with each other) so that i points to the east, and j to the north, k will point upward. When rectangular axes of X, Y, and Z are employed, their directions will be conformed to a similar con¬ dition, and i, j , k (when the contrary is not stated) will be supposed parallel to these axes respectively. We may have occasion to use more than one such system of unit vectors, just as we may use more than one system of coordinate axes. In such cases, the different systems may be distinguished by accents or otherwise. 12. Numerical computation of a geometrical sum. —If p —aa + b/3 + cy, < 3 —ci r a + b'fi + c'y, &c., then p-p g -j- &c. — {a -f- a' + &c.j cx.-\- {b -1 - b +&c.)/?+(c + c +&c i)y. I. e., the coefficients by which a geometrical sum is expressed in terms of three vectors are the sums of the coefficients by which the separate terms of the geometrical sum are expressed in terms of the same three vectors. VECTOR ANALYSIS. 5 Direct and Shew Products of Vectors. 13. Def —The direct product of a and ft (written a. ft) is the scalar quantity obtained by multiplying the product of their magnitudes, by the cosine of the angle made by their direc¬ tions. 14. Def .—The skew product of a and ft (written axft) is a vector function of a. and ft. Its magnitude is obtained by multiplying the product of the magnitudes of a. and /3 by the sine of the angle made by their directions. Its direction is at right angles to a. and ft, and on that side of the plane contain¬ ing a and ft (supposed drawn from a common origin), on which a rotation from o .: to ft through an arc of less than 180° appears counter-clock-wise. The direction of aXft may also be defined as that in which an ordinary screw advances as it turns so as to carry a : toward ft. Again, if a be directed toward the east, and ft lie in the same horizontal plane and on the north side of a, aX ft will be directed upward. 15. It is evident from the preceding definitions that a.13=13.a, and aX [3—— [3Xoi. 16. Moreover, [gn]. (3 = a. [n/3] = n[a. /3 ], and [na]x/3—aX[nf3]=n[aXf3]. The brackets may therefore be omitted in such expressions. 17. From the definitions of Mo. 11 it appears that i. i —j.j—k .k= 1, i.j =j. i=i. k — k. i =j. k=k.j = 0, iXi= 0, jXj= 0, kxk= 0, i Xj=k, jXk=i , k X i =j, jXi——k , k xj= — q i X, k= —j. 18. If we resolve ft into two components ft' and ft", of which the first is parallel and the second perpendicular to a, we shall have a.j3 = a.f3’ and aX f3=ot.X j3". 19. a.[/3 + y] = a./3+a.y and aX[/3 + y] = aX/3 + aXy. To prove this, let a—ft+y, and resolve each of the vec¬ tors ft, y, a into two components, one parallel and the other perpendicular to a. Let these be ft', ft", y', y", a', a". Then the equations to be proved will reduce by the last section to a . a'—a . ft + a . y' and <xXa"=aX f3" + aXy". 6 VECTOR ANALYSIS. Now since a=ap s jT we may form a triangle in space, the sides of which shall be ft, y, and a. Projecting this on a plane per¬ pendicular to a, we obtain a triangle having the sides ft", y", and a", which affords the relation a"—ft !, -\-y". If we pass planes perpendicnlar to a through the vertices of the first triangle, they will give on a line parallel to a segments eqnal to ft',y',a\ Thus we obtain the relation a'=ft ; +y ; . There¬ fore a. 0 '=a.ft'+a.y', since all the cosines involved in these pro¬ ducts are equal to unity. Moreover, if a is a unit vector, we shall evidently have a X a"= aX ft" + a X y", since the effect of the skew multiplication by a upon vectors in a plane perpen¬ dicular to a is simply to rotate them all 90° in that plane. But any case may be reduced to this by dividing both sides of the equation to he proved by the magnitude of a. The proposi¬ tions are therefore proved. 20. Hence, [n + /t] . y = a.y + ft.y, [a + ft] X y=aXy + ftX y, [a' + /T] . {y-\-d^] = a.y-^(.d~\-ft.y + ft.6, [a + /3\x[y+$]=aXy + a^S + fiXy + fiXd; and, in general, direct and skew products of sums of vectors may be expanded precisely as the products of sums in algebra, except that in skew products the order of the factors must not he changed without compensation in the sign of the term. If any of the terms in the factors have negative signs, the signs of the expanded product (when there is no change in the order of the factors), will he determined by the same rules as in algebra. It is on account of this analogy with algebraic prod¬ ucts that these functions of vectors are called products and that other terms relating to multiplication are applied to them. 21. Numerical, calculation of direct and skew products — The properties demonstrated in the last two paragraphs (which may be briefly expressed by saying that the operations of direct and skew multiplication are distributive) afford the rule for the numerical calculation of a direct product, or of the components of a skew product, when the rectangular compo¬ nents of the factors are given numerically. In fact, if a—xi + yj+zk , and fi=x r i+y'j+ztk ; a. ft—xx' + yy' + zz r , and ax ft={yd—zy')i + (zx'—xz')j + (xy' — yx')k. 22 . Representation of the area of a parallelogram by a skew products. —It will be easily seen that a X ft represents in magnitude the area of the parallelogram of which a and ft (sup¬ posed drawn from a common origin) are the sides, and that it represents in direction the normal to the plane of the parallel- VECTOR ANALYSIS. 1 ogram on the side on which the rotation from a toward appears counter-clock-wise. 23. Representation of the volume of a parallelopiped by a triple product .—It will also be seen that aXfi.f represents in numerical yalue the volume of the parallelopiped of which a, /9, and y (supposed drawn from a common origin), are the edges, and that the value of the expression is positive or nega¬ tive according as y lies on the side of the plane of a and /9 on which the rotation from a. to /9 appears counter-clock-wise, or on the opposite side. 24. Hence, aX/3.y=/3Xy.a=yXaf=y-aX/3=a./3xy —fi.yXot— —ft X a.y= — y X fi.a= — a X y.fi = —y.pXa=—a.yXft=—fi.aXy. It will he observed that all the products of this type, which can he made with three given vectors, are the same in numer¬ ical value, and that any two such products are of the same or opposite character in respect to sign, according as the cyclic order of the letters is the same or different. The product van¬ ishes when two of the vectors are parallel to the same line, or when the three are parallel to the same plane. This kind of product may be called the scalar product of the three vectors. There are two other kinds of products of three vectors, both of which are vectors, viz: products of the type ( a.ffy or y (a . /9), and products of the type ax{fXy] or \jXff\Xa. 25. i.jX k—j.kx i—k.iXj= 1. i.kxj—k.j X i=jiXk= — 1. ' From these equations, which follow immediately from those of No. 17, the propositions of the last section might have been derived, viz: by substituting for a, /9, and y, respectively, expressions of the form xi+yj+zk, x'i+y'j+z'k, and x"i~\-y"j +z"kf Such a method, which may be called expansion in terms of i, j, and k, will on many occasions afford very simple, although perhaps lengthy, demonstrations. 26. Triple products containing only two different letters .— The significance and the relations of (a. a)/9, (a.fa, and aX [«X/9] will be most evident, if we consider /9 as made up of * Since the sign x is only used between vectors, the skew multiplication in expressions of this kind is evidently to be pel-formed first. In other words, the above expression must be interpreted as [a x /?] . y. f The student who is familiar with the nature of determinants will not fail to observe that the triple product a.f3 x y is the determinant formed by the nine rectangular components of a, (3, and y, nor that the rectangular components of a x (3 are determinants of the second order formed from the components of a and f3. (See the last equation of No. 21.) 8 VECTOR ANALYSIS. two components, f and B'\ respectively parallel and perpen¬ dicular to a. Then fi—fi'f- (3", ( a. (3) a=[a.(3') a=[a.a) (3\ aX[&X fi] = ax[ax /3 r '] = — (a. a) (3". Hence, ffX [a X (3~\=d[a.(3) a—(a.a) (3. 27. General relation of the vector products of three factors . —In the triple product « X [/? X y] we may set a=l/3 + my + n/3xy , unless /9 and y have the same direction. Then ax[/3xy]=lfix[/3xy]+myx[/3xy] —l {fi.y) f3—l(/3./3) y—m (y.fi) y + m (y.y) (3 ■=(l/3.y+my.y) )3 — (l/3.(3 + my.f3 ) y. But lf3.y + my.y=a.y, and 1(3.(3my. (3—a. (3. VoiVjly -y.nfr Therefore ax[(3Xy] = (a.y) (3— [a.(3) y, which is evidently true, when {3 and y have the same directions. It may also be written [yX/3] Xa=P (y.a) — y {(3.a). 28. This principle may he used in the transformation of more complex products. It will he observed that its applica¬ tion will always simultaneously eliminate, or introduce, two signs of skew multiplication. The student will easily prove the following identical equa¬ tions, which, although of considerable importance, are here given principally as exercises in the application of the preced¬ ing formulae. 29. aX[/3xy]+/3X[yX<x} + yX[aX/3]= 0. 30. [ax(3].[yX6] = (a.y) (/3.6) — (a.6) (/lyf 31. [aX/3]X[yXS]—(a.yXd) (3— {fi.yx 6) a = (a./3x<3) y—{a.(3xy) d. s 32. axipX[yXd]]=(a.yXd) (3—[a.(3) yX6 — [(3.6) axy—{/3.y ) «Xd. 33. [aX/3]. [yX6]x[£XZ]=(a-ftx6){y.£XZ)—(a./3xy)(6.ex£) = (a.f3x Z) fy X 6)-\-[a.(3x e) (<g. y X d) = {y.dxa)[(3.exd) — {y-6x(3)(a.exd). 34. [aXft]. \_(3Xy\X\yXoi\ — (a.f3XyY' 11. i avfhy -4- Vy v&p ~ & 3d. &-y. — oiS.fiy y. — 3l. W<ip>\/Y$ — ^ \/y§ cs 3*. VvVfiVy? = /3.0tK r <f_ XuV/iy ~ V<*y — fy. 33. v «fiVY f 2Vzt ) ~ < V(iS'.- r Ve$ - C iV/iy. JVcJ V Voi/iV Y/i r V r u = YECTOE ANALYSIS. 9 35. The student will also easily convince himself that a product formed of any number of letters (representing vectors) combined in any possible way by scalar, direct, and skew mul¬ tiplications may be reduced by the principles of Nos. 24 and 27 to a sum of products, each of which consists of scalar fac¬ tors of the forms a. ft and a.ftxy, with a single vector factor of the form a or a X ft, when the original product is a vector. 36. Elimination of scalars from rector eolations . — It has already been observed that the elimination of vectors from ecpiations of the form aa + bfi + cy + d S + &c. = 0 is performed by the same rule as the eliminations of ordinary algebra. (See No. 9.) But the elimination of scalars from such equations is at least formally different. Since a single vector equation is the equivalent of three scalar equations, we must be able to deduce from such an equation a scalar equa¬ tion from which two of the scalars which appear in the orig¬ inal vector equation have been eliminated. We shall see how this may be clone, if we consider the scalar equation a a. X + b/3. X + cy. X + cl 6. X + &c. == 0, which is derived from the above vector equation by direct mul¬ tiplication by a vector X. We may regard the original equa¬ tion as the equivalent of the three scalar equations obtained by substituting for a, ft, y, d, etc., their X-, Y-, and Z- compo¬ nents. The second equation would be derived from these by multiplying them respectively by the X-, Y-, and Z- compo¬ nents of X and adding. Hence the second equation may be regarded as the most general form of a scalar equation of the first degree in a, b, c, d, etc., which can be derived from the original vector equation or its equivalent three scalar equations. If we wish to have two of the scalars, as b and c, disappear, we have only to choose for X a vector perpendicular to ft and y. Such a vector is ftXy. We thus obtain aa.fi Xy+ d6.fi x y 4- &c.=0. 37. Relations of four rectors. —By this method of elimina¬ tion we may find the values of the coefficients a, b, and c in the equation p=aa-\-bfi + cy, (1) by which any vector p is expressed in terms of three others. (See No. 10.) If we multiply directly by ft X y, y X a, and axft, we obtain p.fiXy=aa.fixy, p.yXa=bfi.yXcv, p.aX fi—cy.aX fi ; (2) whence 2 10 VECTOR, ANALYSIS. a=M>< T; b= RZl If P-<*XP at.fiXy 9 a.ftxy’ a.fixy ^ ' By substitution of these values, we obtain the identical equa¬ tion, (a.fixy) p=(p.fixy) a + (p.yx<x) fi+(p.axfi) y ( 4 ) (Compare Ho. 31.) If we wish the four vectors to appear symmetrically in the equation we may write {o'.fSxy) p—{/3.yXp) a + (y.pXa) /3—(p.axW ) y = 0. ( 5 ) If we wish to express p as a sum of vectors having directions perpendicular to the planes of a and ft, of ft and y, and of y and a, we may write P=efiXy+fyXa+gax/3. ( 6 ) To obtain the values of e,f g, we multiply directly by a, by ft, and by y. This gives e — _ P’ a f— _ P -fi r/= _ p -y ft.yXoi ' y.aXfP J at.fiXy (1) Substituting these values we obtain the identical equation (a.ftXy) p={p.a) fiXy+(p.fi) yX<x+(p.y) acXfi. (8) (Compare Ho. 32.) 38. Reciprocal systems of rectors .—The results of the pre¬ ceding section may be more compactly expressed if we use the abbreviations a ./_ fixy' yxa axfi ac.fi Xy* fi.yXoi ' y.aXfi' 0) The identical equations (4) and (8) of the preceding number thus become p=(p.a) a+(p.fi') fi+(p.y') y, p=(p.a) a'+(p.fi ) fi'+(p.y) y'. (2)) (3) We may infer from the similarity of these equations that the relations of «, ft, y, and o!, ft', y' are reciprocal; a proposition which is easily proved directly. For the equations _ fi'xy o y'Xoc' _ a'xfi’ a ~a’.p'xy” 1 P'.y'Xa” r ~y',a'xP' are satisfied identically by the substitution of the values of o!, ft ', and y' given in equations (1). (See Hos. 31 and 34.) Ref .—It will be convenient to use the term reciprocal to designate these relations, i. e., we shall say that three vectors are reciprocals of three others, when they satisfy relations sim¬ ilar to those expressed in equations (1) or (4). ph aV/4, VAC rvpy r " I ary'* I VECTOR ANALYSIS. 11 With this understanding we may say:— The coefficients hy which any vector is expressed in terms of three other vectors are the direct products of that vector with the reciprocals of the three. Among other relations which are satisfied hy reciprocal sys¬ tems of vectors are the following: a.a'—fi.fi’ — y.y' z=l\. [a.fixy) = •(See No. 34.) aXot' + fiXfi' + y Xy'=o. (See No. 29.) A system of three mutually perpendicular unit reciprocal to itself, and only such a system. The identical equation ( 5 ) =/ (») vectors is s ^ P= (P-i) i + (P-j) j+ (p.Je) k may be regarded as a particular case of equation (2). The system reciprocal to a.X[ 3, fi X y, yXa is a fi fi a. fix y* oc.fi xy' -a.fixy t ( 8 ) ?= L.if+j.ptrU^ Vfa. 39. Scalar equations of the first degree with respect to unknown rector. —It is easily shown that any scalar equation of the first degree with respect to an unknown vector p 9 in which all the other quantities are known, may he reduced to the form p.ac=za, (Ayi Y,y Y al _ y tW aA o an / # in which a. and a are known. (See ISTo. 35.) Three such equations will afford the value of p (hy equation (8) of No. 37, or equation (3) of No. 38), which may he used to eliminate p from any other equation either scalar or vector. When we have four scalar equations of the first degree with respect to p, the elimination may he performed most symmet¬ rically hy substituting the values of p.a etc., in the equation, (p.oc) (fi.yX 8 ) — (p.fi) (y.Sxoc)+ (p.y) (d.ax fi) — (p.d) ( a.fixy ), which is obtained from equation (8) of No. 37 hy multiplying directly hy d. It may also be obtained from equation (5) of No. 37 hy writing a for p, and then multiplying directly by p. 40. Solution of a rector equation of the first degree with respect to the unknown rector .—It is now easy to solve an equation of the form 8—a (k.p) + fi (p.p) + y (v.p), ~ c4 - Xy -y f 2 "- Yjfi ~ +-/5-r + ^.' ? )p f - ««'+ 0+ v.y) S' n.fi +. vqq +p.f- rfy-v) ( 1 ) err- 2 = fp i 12 VECTOR ANALYSIS. where «, /9, y, d , A, p, and v represent known vectors. Multi¬ plying directly by /?Xp, by yX«, and by axft we obtain fi.yX8=(f3.yXoi) (A.p), yaX 8=(y.aXfi) (p.p), a.fix8 = {a.fixy) (u.p); or a.6—\.p, fi'.d—jj.p, y'.8=y.p, where a 1 , f, y' are the reciprocals of a, j3, y. Substituting these values in the identical equation p—A'/A.p) 4 - p'(p.p) 4- r'(y.p ), in which A', p', v' are the reciprocals of A, p, v, (see Ho. 38,) we have p=V(a |d) + p/(/3'.d) 4 (2) which is the solution required. It results from the principle stated in Ho. 35, that any vector equation of the first degree with respect to p may be reduced to the form 8 = p(A.p) + [8 (p.p) 4- y{y.p) + ap 4 s x p. But ap—oK! (A.p) 4«p'(p.p) + ar'(r.p ), and £x p=£X A'(A.p) + fX p'(p-p) + sX y'(v.p ), a.(XV+-) P where A', //, i/ represent, as before, the reciprocals of A, p, v. f By substitution of these values the equation is reduced to the - Vq\ . A p form of equation (1), which may therefore be regarded as the 4-..« * most general form of a vector equation of the first degree with respect to p. 41. Relations between two normal systems of unit rectors .— If i, f A, and i/, k! are two normal systems of unit vectors, we have a — (i.i f )i 4 {j.i' )j+ ( Je.i ' )k, j f = (if )i + 07 )j+ ( k.f )k, | ( 1 ) k'=(i.k')i + (j.k')j 4 (k.k')k, ) and i = ( i.i ) i' + ( ij’lf + ( l/. J ')/■:, i j=(j.iy + (jj')j' + (j.k')k',\ (2) k = (k.i')% 4 (k.j )j' 4 ( k.k')k'. ) (See equation 8 of Ho. 38.) The nine coefficients in these equations are evidently the cosines of the nine angles made by a vector of one system with a vector of the other system. The principal relations of these cosines are easily deduced. By direct multiplication of each of the preceding equations with itself, we obtain six equations of the type (i.i'Y + (j.ir + (k.ir=l- ( 3 ) 4- 4. VECTOR ANALYSIS. 13 By direct multiplication of equations (1) with each other, and of equations (2) with each other, we obtain six of the type (W) (if) + (j.i') (j.jj + (hi') (hf) = 0. (4) By skew multiplication of equations (1) with each other, we obtain three of the type k'={(j.ij (hf)-(hi') I (hi') (i.j')-(i.i') hjj }j + {(bb) (j-j')-(j-i') (ij')}h Comparing these three equations with the original three, we obtain nine of the type iM=(j.i')(hf)-(hi')(jf). (5) Finally, if we equate the scalar product of the three right hand members of (1) with that of the three left hand members, we obtain (U') 07) (hh) + (if) (j.h) (hi') + (i.h) (j.i') (kf) -(hi') 07) (i.A/)-(V) tw (M)-(hh) (j.i') 07)=1. ( 6 ) Equations (1) and (2) (if the expressions in the parentheses are supposed replaced by numerical values) represent the linear relations which subsist between one vector of one system and the three vectors of the other system. If we desire to express the similar relations which subsist between two vectors of one system and two of the other, we may take the skew products of equations (1) with equations (2), after transposing all terms in the latter. This will afford nine equations of the type (if)h-(i.h)f=(hi')j-(j.i')h ( 7 ) 14 VECTOR ANALYSIS. CHAPTER II. CONCERNING THE DIFFERENTIAL AND INTEGRAL CALCULUS OF VECTORS. 42. Differentials of vectors .—The differential of a vector is the geometrical difference of two values of that vector which differ infinitely little. It is itself a vector, and may make any angle with the vector differentiated. It is expressed by the same sign iff) as the differentials of ordinary analysis. With reference to any fixed axes, the components of the differential of a vector are manifestly equal to the differentials of the components of the vector, i. e., if «, /3, and y are fixed unit vectors, and p=xa + yfi+zy, dp=dx a J r dyfj + dz y. 43. Differential of a function of several variables .—The differential of a vector or scalar function of any number of vector or scalar variables is evidently the sum (geometrical or algebraic, according as the f unction is vector or scalar,) of the differentials of the function due to the separate variation of the several variables. 44. Differential of a product .—The differential of a product of any kind due to the variation of a single factor is obtained by prefixing the sign of differentiation to that factor in the product. This is evidently true of differentials, since it will hold true even of finite differences. 45. From these principles we obtain the following identical equations: d[a -f- fp —^ da -|- dfi^ (l) d(na)=dn a + n da , (2) d(a. j3)— da. ft-y a.dfi, (3) d[aX fi]=daX +aXdfi, (4) d(a.fix y)= da.fi Xy + a. dp Xy + a.ft X dy, (5) d[(a.fi)y] = (da.fi)y+ (a.dfi)y + (a.fi)dy. (6) 46. Differential coefficient with respect to a scalar .—The quotient obtained by dividing the differential of a vector due to the variation of any scalar of which it is a function by the differential of that scalar is called the differential coefficient of the vector with respect to the scalar, and is indicated in the same manner as the differential coefficients of ordinary analysis. VECTOR ANALYSIS. 15 If we suppose the quantities occurring in the six equations of the last section to he functions of a scalar t, we may substi¬ tute — for d in those equations since this is only to divide all Cllr terms by the scalar dt. 41. Successive differentiations .—The differential coefficient of a vector with respect to a scalar is of course a finite vector, of which we may take the differential, or the differential coef¬ ficient with respect to the same or any other scalar. We thus obtain differential coefficients of the higher orders, which are indicated as in the scalar calculus. A few examples will serve for illustration. If p is the vector drawn from a fixed origin to a moving point at any time t, y will he the vector representing the (JjTj ddp velocity of the point, and the vector representing its accel- (lu eration. If p is the vector drawn from a fixed origin to any point on a curve, and s the distance of that point measured on the (7 Q curve from any fixed point, is a unit vector, tangent to the CtS • ^ Q curve and having the direction in which s increases: —- is a ds vector directed from a point on the curve to the center of curv¬ ature, and equal to the curvature: ~ X is the normal to the ds ds osculating plane, directed to the side on which the curve appears described counter-clock-wise about the center of curvature, and equal to the curvature. The tortuosity (or rate of rotation of the osculating plane, considered as positive when the rota¬ tion appears counter-clock-wise as seen from the direction in which s increases,) is represented by dp d 3 p d 3 p ds * ds 2 ^ ds 3 dp df> ds' 1 -* ds ^ 48. Integration of an equation between differentials .—If t and u are two single-valued continuous scalar functions of any number of scalar or vector variables, and then dt—du , t — u -f- a. where a is a scalar constant. 16 VECTOR ANALYSIS. Or, if t and co are two single-valued continuous vector func¬ tions of any number of scalar or vector variables, and dr—dcsD , then t=go+o', where a is a vector constant. "When the above hypotheses are not satisfied in general, but will be satisfied if the variations of the independent variables are confined within certain limits, then the conclusions will hold within those limits, provided that we can pass by continu¬ ous variation of the independent variables from any values within the limits to any other values within them, without transgressing the limits. 49. So far, it will be observed, all operations have been entirely analogous to those of the ordinary calculus. Functions of Position in Space. 50. Pef. —If u is any scalar function of position in space, (i. e., any scalar quantity having continuously varying values in space,) pu is the vector function of position in space which has everywhere the direction of the most rapid increase of u, and a magnitude equal to the rate of that increase per unit of length, pu may be called the derivative of u, and u, the primitive of pu. We may also take any one of the bTos. 51, 52, 53 for the definition of pu. 51. If p is the vector defining the position of a point in space, du—pu.dp. 52. 53. .du .du 7 du r u=l 7x +J 7j +k dz du dx —i.pu , du . du 7 p z =lc F u - dy 54 . Def. —If co is a vector having continuously varying values in space, . doo . dco dco , < 7 - 6,= f* + ^ +i -&’ (1) . da) . da) dco and f 7x<a =* x ^ + ' ?x ^ +/l ' x &- p.co is called the divergence of co and p X co its curl. If we set rc)=Xi + Yj +Z/q VECTOB ANALYSIS. 11 we obtain by substitution tbe equations dX d,Y dZ 00 dx dy ^ dz' , •/ and /7 X go— ^( V dZ dy dX\ dyr wbicli may also be regarded as defining p.co and y X co. 55. Surface-integrals .—The integral ffco.da , in which da represents an element of some surface, is called the surface- integral of co for that surface. It is understood here and else¬ where, when a vector is said to represent a plane surface, (or an element of surface, which may be regarded as plane,) that the magnitude of the vector represents the area of the surface, and that the direction of the vector represents that of the nor¬ mal drawn toward the positive side of the surface. When the surface is defined as the boundary of a certain space, the out¬ side of the 'surface is regarded as positive. The surface-integral of any given space (i. e., the surface- integral of the surface bounding that space) is evidently equal to the sum of the surface-integrals of all the parts into which the original space may be divided. Tor the integrals relating to the surfaces dividing the parts will evidently cancel in such a sum. The surface-integral of to for a closed surface bounding a space dr infinitely small in all its dimensions is J7. GO dv. This follows immediately from the definition of yco, when the space is a parallelopiped bounded by planes perpendicular to i, j, k. In other cases, we may imagine the space—-or rather a space nearly coincident with the given space and of the same volume dv —to be divided up into such parallelopipeds. The surface-integral for the space made up of the parallelopipeds will be the sum of the surface-integrals of all the parallelo¬ pipeds, and will therefore be expressed by p.co dv. The sur¬ face-integral of the original space will have sensibly the same value, and will therefore be represented by the same formula. It follows that the value of p.co does not depend upon the system of unit vectors employed in its definition. It is possible to attribute such a physical signification to the quantities concerned in the above proposition, as shall make it evident almost without demonstration. Let us suppose co to represent a flux of any substance. The rate of decrease of the density of that substance at any point will be obtained by dividing the surface-integral of the flux for any infinitely small closed surface about the point by the volume enclosed, 3 18 VECTOR ANALYSIS. This quotient must therefore he independent of the form of the surface. We may define p.ta as representing that quotient, and then obtain equation (1) of bio. 54 by applying the general principle to the case of the rectangular parallelopiped. 56. Shew surface-integrals. —The integral fjdaXto maybe called the skew surface-integral of to. It is evidently a vector. Tor a closed surface hounding a space dr infinitely small in all dimensions, this integral reduces to pXtodv, as is easily shown by reasoning like that of bio. 55. 51. Integration .—If dr represents an element of any space, and da an element of the hounding surface, ff/V • dv =ff a) - d <?• For the first member of this equation represents the sum of the surface integrals of all the elements of the given space. We may regard this principle as affording a means of integration, since we may use it to reduce a triple integral (of a certain form) to a double integral. The principle may also he expressed as follows: The surface-integral of any vector function of position in space for a closed surface is equal to the volume-integral of the divergence of that function for the space enclosed. 58. Line-integrals .—The integral J'to.dp, in which dp de¬ notes the element of a line, is called the line-integral of to for that line. It is implied that one of the directions of the line is distinguished as positive. When the line is regarded as hounding a surface, that side of the surface will always he regarded as positive, on which the surface appears to he cir¬ cumscribed counter-clock-wi se. 59. Integration. —From JNo. 51 we obtain directly fyu .dp—y!'—u where the single and double accents distinguish the values relating to the beginning and end of the line. In other words,—The line-integral of the derivative of any (continuous) scalar function of position in space is equal to the difference of the values of the function at the extremities of the line. For a closed line the integral vanishes. 60. Integration. —The following principle may he used to reduce double integrals of a certain form to simple integrals. If da represents an element of any surface, and dp an element of the hounding line, ffV X oo.d,(i—f oo.dp. In other words,—The line-integral of any vector function of position in space for a closed line is equal to the surface-inte- VECTOR ANALYSIS. 19 gral of the curl of that function for any surface hounded by the line. To prove this principle, we will consider the variation of the line-integral which is due to a variation in the closed line for which the integral is taken. We have, in the first place, 6/oo.dp=/S oo.dp+f go. 6 dp. But QD.ddp—d{GD.S p)—doo.d p. Therefore, since fdico.dp)— 0 for a closed line, Now and 6foo.dp=-fdGD.dp—fdGD.d p. 1 8 1 doo |_*d /J) j doo=J> cl GO , dx dx = 2 'doo dx {t - dp) ’ where the summation relates to the coordinate axes and con¬ nected quantities. Substituting these values in the preceding equation, we get Sf,-W.dp=f2((i.dp) - (Up)(^. Sp^j, or by Ho. 30, 5/GD.dp—f2 d(sD dx . [dp X dp] =fp X go. [dp X dp]. But dpxdp represents an element of the surface generated by the motion of the element dp, and the last member of the equation is the surface-integral of j 7 Xco for the infinitesimal surface generated by the motion of the whole line. Hence, if we conceive of a closed curve passing gradually from an infinitesimal loop to any finite form, the differential of the line- integral of co for that curve will be equal to the differential of the surface integral of y X co for the surface generated: therefore, since both integrals commence with the value zero, they must always be equal to each other. Such a mode of generation will evidently apply to any surface closing any loop. 61. The line-integral of co for a closed fine bounding a plane surface da infinitely small in all its dimensions is therefore j7 X go. da. This principle affords a definition of pXco which is inde¬ pendent of any reference to coordinate axes. If we imagine a circle described about a fixed point to vary its orientation while keeping the same size, there will be a certain position of the circle for which the line-integral of co will be a maximum, unless the line-integral vanishes for all positions of the circle. The axis of the circle in this position, drawn toward the side 20 VECTOR ANALYSIS. on which a positi ve motion in the circle appears counter-clock¬ wise, gives the direction of y X co, and the quotient of the inte¬ gral divided by the area of the circle gives the magnitude of /7 X co. y, y., and (7 X allied to Functions of Functions of Position. 62. A constant scalar factor after y, y., or y x may be placed before the symbol. 63. If f(u) denotes any scalar function of u, and fO l) the derived function, Cf(f)=f {u)yu. 64. If u or co is a function of several scalar or vector varia¬ bles, which are themselves functions of the position of a single point, the value of yu or y.co or yXco will be equal to the sum of the values obtained by making successively all but each one of these variables constant. 65. By the use of this principle, we easily derive the follow¬ ing identical equations: y(t + U ) = yt+ yU. (1) y.(r + oo) = y.r + y.Go. yX[r + co] = yXr + yXGo. (2) y(tu)=uyt + tyv. (3) y.(uGo) = CsD.yu-\-v,y.Go. (4) y X \u go] = ny Xgo—goX yu. (5) y.\rXGo] — co.yX'r—r.yXco. (6) The student will observe an analogy between these equations and the formula} of multiplication. (In the last four equations the analogy appears most distinctly when we regard all the fac¬ tors but one as constant.) Some of the more curious features of this analogy are due to the fact that the y contains implic¬ itly the vectors i, j, and k, which are to be multiplied into the following quantities. Combinations of the Operators y, y., and y X . 66. If u is any scalar function of position in space, yxyu= o, as may be derived directly from the definitions of these ope¬ rators. 6T. Conversely, if co is such a vector function of position in space that yXGO — 0, VECTOR ANALYSIS. 21 co is the derivative of a scalar function of position in space. This will appear from the following considerations: The line-integral fco.dp will vanish for any closed line, since it may he expressed as the surface-integral of p X co. (No. 60.) The line-integral taken from one given point P / to another given point P 77 is independent of the line between the points for which the integral is taken. (For, if two lines joining the same points gave different values, by reversing one we should obtain a closed line for which the integral would not vanish.) If we set u equal to this line-integral, supposing P 77 to be variable "and P 7 to be constant in position, u will be a scalar function of the position of the point P 77 , satisfying the condi¬ tion du=co.dp, or, by No. 51, pu—co. There will evidently be an infinite number of functions satisfying this condition, which will differ from one. another by constant quantities. If the region for which pXco — 0 is unlimited, these func¬ tions will be single-valued. If the region is limited, but acyclic,* the functions will still be single-valued and satisfy the condition pu=co within the same region. If the region is cyclic, we may determine functions satisfying the condition p'u—co within the region, but they will not necessarily be single-valued. 68. If co is any vector function of position in space, p.pXco= 0. This may be deduced directly from the defini¬ tions of No. 54. The converse of this proposition will be proved hereafter. 69. If u is any scalar function of position in space, we have by ISTos. 52 and 54 (d* d 2 cV\ r-r v= W + ty + **r TO. Def . — If co is any vector function of position in space, we may define p.pco by the equation / cf d* cl 2 \ ^= 1 *XThf + dsN * If every closed line within a given region can contract to a single* point without breaking its continuity, or passing out of the region, the region is called acyclic, otherwise cyclic. A cyclic region may be made acjmlic by diaphragms, which must then be re¬ garded as forming part of the surface bounding the region, each diaphragm contributing its own area twice to that surface. This process may be used to reduce many-valued functions of position in space, having single-valued deriva¬ tives, to single-valued functions. When functions are mentioned or implied in the notation, the reader will always understand single-valued functions, unless the contrary is distinctly intimated, or the case is one in which the distinction is obviously immaterial. Diaphragms may be applied to bring functions naturally many-valued under the application of some of the following theorems, as Nos. 14 £f. 22 VECTOR ANALYSIS. the expression p.p being regarded, for the present at least, as a single operator when applied to a vector. (It will he remem¬ bered that no meaning has been attributed to p before a vec¬ tor.) It should be noticed that, if ao=iX+jY + kZ, fcpGo=ip.pX+jp.prY+kp.pZ, that is, the operator p.p applied to a vector affects separately its scalar components. 71. From the above definition with those of Flos. 52 and 54 we may easily obtain V- V FF * 00 ~ V X v X The effect of the operator p.p is therefore independent of the directions of the axes used in its definition. 72. The expression — {a^p.pu, where a is any infinitesimal scalar, evidently represents the excess of the value of the scalar function u at the point considered above the average of its values at six points at the following vector distances: ai, — ai, aj, — aj, ak, — ak. Since the directions of i, j, and k are immaterial, (provided that they are at right angles to each other), the excess of the value of u at the central! point above its average value in a spherical surface of radius a constructed about that point as the center will be represented by the same expression, — }a 2 p.pit. Precisely the same is true of a vector function, if it is un¬ derstood that the additions and subtractions implied in the terms average and excess are geometrical additions and sub¬ tractions. Maxwell has called — p.pw the concentration of u, whether u is scalar or vector. We may call p.pw (or p.po), which is proportioned to the excess of the average value of the func¬ tion in an infinitesimal spherical surface above the value at the center, the dispersion of u (or co). Transformation of Definite Integrals. 73. From the equations of Ho. 65, with the principles of integration of Hos. 57, 59, and 60, we may deduce various transformations of definite integrals, which are entirely analo¬ gous to those known in the scalar calculus under the name of integration by parts. The following formulae (like those of Hos. 57, 59, and 60) are written for the case of continuous values of the quantities (scalar and vector) to which the signs p. p., and p X are applied. It is left to the student to complete the formulae for cases of discontinuity in these values. The manner in which this is to be done may in each case be inferred VECTOR ANALYSIS. 23 from the nature of the formula itself. The most important discontinuities of scalars are those which occur at surfaces: in the case of vectors, discontinuities at surfaces, at lines, and at points, should he considered. 74. From equation (3) we obtain f 7 (tu) .dp=t"u" — t'u' —fupt.dp 4- ftpu.dp , where the accents distinguish the quantities relating to the limits of the line-integrals. We are thus able to reduce a line-integral of the form fupt.dp to the form — ftpu.dp with quantities free from the sign of integration. 75. From equation (5) we obtain ffp X (u go). do —Ju gj. dp —ffupX oo.da —ffGoXpu.dp , where, as elsewhere in these equations, the line-integral relates to the boundary of the surface integral. From this, by substitution of pt for to, we may derive as a particular case ffpux pt.dff=fupt.dp= —ftpu.dp. 76. From equation (4) we obtain Iff (7- \ugd\ dv =/Juoo.da =fff go. pu dv + fffup. go dv, where, as elsewhere in these equations, the surface-integral relates to the boundary of the volume-integrals. From this, by substitution of pt for to, we derive as a partic¬ ular case fffpt.pu dv^ffupt.do-fffup. V tdv =jftpu.dG-ffftp.pu dv, which is Green’s Theorem. The substitution of spt for to gives the more general form of this theorem which is due to Thomson, viz:— fffspt.pu dv —ffusp t.dtr —fffu p.[spt]dv —ff^pu.da—ffftp. [spu]dv. 77. From equation (6) we obtain Jlfv\ r X oo]dv=ffr X Go.dG=fffoo.pX r dv-fffr.p X go dv. A particular case is P X oo dv —ffoo x pu.dtJ. Integration of Differential Equations. 78. If throughout any continuous space (or in all space) pu—0, 24 VECTOR ANALYSIS. then throughout the same space u— constant. 79. If throughout any continuous space (or in all space) / 7 . /7 'V> —■ 0 , and in any finite part of that space, or in any finite surface in or bounding it, pu= 0, then throughout the whole space yu— 0, and u— constant. This will appear from the following considerations. If yn~ 0 in any finite part of the space, u is constant in that part. If u is not constant throughout, let us imagine a sphere situated principally in the part in which u is constant, hut pro¬ jecting slightly into a part in which u has a greater value, or else into a part in which u has a less. The surface-integral of yu for the part of the spherical surface in the region where u is constant will have the value zero : for the other part of the surface, the integral will be either greater than zero, or less than zero. Therefore the whole surface-integral for the spher¬ ical surface will not have the value zero, which is required by the general condition, p.p’-u— 0. Again, if yu =0 only in a surface in or bounding the space in which y.yu — 0, u will be constant in this surface, and the surface will be contiguous to a region in which y.yu —0 and u has a greater value than in the surface, or else a less value than in the surface. Let us imagine a sphere lying principally on the other side of the surface, but projecting slightly into this region, and let us particularly consider the surface-integral of yu for the small segment cut off by the surface yu— 0. The integral for that part of the surface of the segment which con¬ sists of part of the surface yu —0 will have the value zero, the integral for the spherical part will have a value either greater than zero or else less than zero. Therefore the integral for the whole surface of the segment cannot have the value zero, which is demanded by the general condition, r-ru=0. 80. If throughout' a certain space (which need not be con¬ tinuous, and which may extend to infinity) y.yu— 0, and in all the bounding surfaces u— constant=«, and (in case the space extends to infinity) if at infinite dist- VECTOR ANALYSIS. 25 ances within the space u=a ,—then throughout the space pu=0, and u—ci. For, if anywhere in the interior of the space pu has a value ditferent from zero, we may find a point P where such is the case, and where u has a value b different from a, —to fix our ideas we will say less. Imagine a surface enclosing all of the space in which u < b. (This must he possible, since that part of the space does not reach to infinity.) The surface-integral of pu for this surface has the value zero in virtue of the general condition p.pic= 0. Put, from the manner in which the surface is defined, no part of the integral can be negative. Therefore no part of the integral can be positive, and the supposition made with respect to the point P is untenable. That the sup¬ position that b > a is untenable may be shown in a similar man¬ ner. Therefore the value of u is constant. This proposition may be generalized by substituting the con¬ dition p.[tpu] = 0 for p.pu=z 0, t denoting any positive (or any negative) scalar function of position in space. The conclusion would be the same, and the demonstration similar. 81. If throughout a certain space (which need not be con¬ tinuous, and which may extend to infinity,) p.pu— 0, and in all the bounding surfaces the normal component of pu vanishes, and at infinite distances within the space (if such there are) r 2 — = 0, where r denotes the distance from a fixed dr origin, then throughout the space pu — 0, and in each continuous portion of the same u= constant. For, if anywhere in the space in question pu has a value different from zero, let it have such a value at a point P, and let u be there equal to b. Imagine a spherical surface about the above-mentioned origin as center, enclosing the point P, and with a radius r. Consider that portion of the space to which the theorem relates which is within the sphere and in which u<db. The surface-integral of pu for this space is equal to zero in virtue of the general condition p.pu— 0. That part of the integral (if any) which relates to a portion of the spherical surface has a value numerically not greater than 4 nr‘ where denotes the greatest numerical value du of — in the portion of the spherical surface considered. 26 VECTOR ANALYSIS. Hence, the value of this part of the surface-integral may be made less (numerically) than any assignable quantity by giving to r a sufficiently great value. Hence, the other part of the surface-integral (viz., that relating to the surface in which u—h, and to the boundary of the space to which the theorem relates,) may be given a value differing from zero by less than any assignable quantity. But no part of the integral relating to this surface can be negative. Therefore no part can be positive, and the supposition relative to the point P is unten¬ able. This proposition also may be generalized by substituting C^'l f $AJj [7.\tf7u] = 0 for f7.pu= 0, and tr 2 — = 0 for r 3 —=0. Chi* - (ajV 82. If throughout any continuous space (or in all space) \71=17U , then throughout the same space t—u-\- const. The truth of this and the three following theorems will be apparent if we consider the difference t—u. 83. If throughout any continuous space (or in all space) f7. pt=p.pu, and in any finite part of that space, or in any finite surface in or bounding it, Vt=pu , then throughout the whole space and t=u + const. 84. If throughout a certain space (which need not be con¬ tinuous, and which may extend to infinity) /7.p2=£7./7W, and in all the bounding surfaces t—u , and at infinite distances within the space (if such there are) t=u , then throughout the space t — U. 85. If throughout a certain space (which need not be con¬ tinuous, and which may extend to infinity) fi', VECTOR ANALYSIS. 27 and in all the bounding surfaces the normal components of ft and fu are equal, and at infinite distances within the space (if such there are) r2 (~^ r ~~ ^^=0, where r denotes the distance from some fixed origin,—then throughout the space ft—fU, and in each continuous part of which the space consists fc*u = constant. 86. If throughout any continuous space (or in all space) fXr—fXoo and f.r = j/.Go, and in any finite part of that space, or in any finite surface in or bounding it, T=GO, then throughout the whole space T— 00. For, since fX(z—co)=0, we may set fu=z—co, making the space acyclic (if necessary) by diaphragms. Then in the whole space u is single-valued and f.fu— 0, and in a part of the space, or in a surface in or bounding it, fu— 0. Flence throughout the space fu—z—co—0. 87. If throughout an aperiphractic* space contained within finite boundaries but not necessarily continuous (7Xi=(7X® and f.r = f.co, and in all the bounding surfaces the tangential components of z and co are equal, then throughout the space T — GO. It is evidently sufficient to prove this proposition for a con¬ tinuous space. Setting fu=z—co , we have f.fu= 0 for the whole space, and u —constant for its boundary, which will be a single surface for a continuous aperiphractic space. ITence throughout the space fu=z—co=0 . 88. If throughout an acyclic space contained within finite boundaries but not necessarily continuous fX z—fX go and f.r = f.GD , and in all the bounding surfaces the normal components of r and co are equal, then throughout the whole space r = go. * If a. space encloses within itself another space, it is called periphradic , other¬ wise aperiphraclic. 28 YECTOE ANALYSIS. Setting f/'U—T—o), we have 0 throughout tlie space, and tlie normal component of pu at tire boundary equal to zero. Hence throughout the whole space pu—T—w= 0. 89. If throughout a certain space (which need not be con¬ tinuous, and which may extend to infinity) p.\7T — \7.\7GQ and in all the bounding surfaces 7—GO , and at infinite distances within the space (if such there are) T=GO, then throughout the whole space r— go. This will be apparent if we consider separately each of the scalar components of r and co. Minimum Yalues of the Volume-integral fjfu co.co dr. {Thomson' 1 s Theorems .) 90. Let it be required to determine for a certain space a vector function of position co subject to certain conditions (to be specified hereafter), so that the volume-integral fffu go. go dv for that space shall have a minimum value, u denoting a given positive scalar function of position. a. In the first place, let the vector co be subject to the con¬ ditions that p.co is given within the space, and that the nor¬ mal component of co is given for the bounding surface. (This component must of course be such that the surface-integral of co shall be equal to the volume-integral fp.codv. If the space is not continuous, this must be true of each continuous portion of it. See No. 57.) The solution is that px(uco)= 0, or more generally, that the line-integral of uco for any closed curve in the space shall vanish. The existence of the minimum requires that fff u go. 8go dv — 0, while do is subject to the limitation that pr.dGO=0, and that the normal component of oco at the bounding surface vanishes. To prove that the line-integral of uco vanishes for VECTOR ANALYSIS. 29 any closed curve within the space, let ns imagine the curve to be surrounded by an infinitely slender tube of normal section dz, which may be either constant or variable. We may satisfy the equation p,dco~ 0 by making dco =0 outside of the tube, and dcodz=da^~ within it, da denoting an arbitrary infinitesimal constant, p the position-vector, and ds an element of the length of the tube or closed curve. We have then fffu go. 6 go dv—fu od.6 go dz ds—fu oo.dp 6a—6a fit Go.dp=0 , whence fu oo.dp= 0 . q. e. d. We may express this result by saying that uco is the derivative of a single-valued scalar function of position in space. (See ISTo. 67.) If for certain parts of the surface the normal component of co is not given for each point, but only the surface-integral of co for each such part, then the above reasoning will apply not only to closed curves, but also to curves commencing and end¬ ing in such a part of the surface. The primitive of uco will then have a constant value in each such part. If the space extends to infinity and there is no special condi¬ tion respecting the value of co at infinite distances, the prim¬ itive of uco will have a constant value at infinite distances within the space or within each separate continuous part of it. If we except those cases in which the problem has no defin¬ ite meaning because the data are such that the integral fuco.codv must be infinite, it is evident that a minimum must always exist, and (on account of the quadratic form of the integral) that it is unique. That the conditions just found are sufficient to insure this minimum, is evident from the consider¬ ation that any allowable values of dco may be made up of such values as we have supposed. Therefore, there will be one and only one vector function of position in space which satisfies these conditions together with those enumerated at the begin¬ ning of this number. b. In the second place, let the vector co be subject to the conditions that /7 X co is given throughout the space, and that the tangential component of co is given at the bounding sur¬ face. The solution is that 17.[u go]=:0, and, if the space is periphractic, that the surface-integral of uco vanishes for each of the bounding surfaces. The existence of the minimum requires that fffu 00.600 dv = 0, % 30 VECTOR ANALYSIS. while ooj is subject to the conditions that (7X<5go= 0 , and that the tangential component of dco in the hounding sur¬ face vanishes. In virtue of these conditions we may set Sgd— f/Sq, where dq is an arbitrary infinitesimal scalar function of posi¬ tion, subject only to the condition that it is constant in each of the bounding surfaces. (See Ho. 67.) By substitution of this value we obtain fffu GD.pSq dv=0, or integrating by parts (Ho. 76) ff u gd.cJij dq—JJ]‘{7.[u Go]dq dv =0. Since dq is arbitrary in the volume-integral, we have through¬ out the whole space ( 7 .[u <»] = 0; and since dq has an arbitrary constant value in each of the bounding surfaces (if the boundary of the space consists of separate parts), we have for each such part ffu Go.dc> — 0 . Potentials , Newtonians , Laplacians. 91. Pef .—If u' is the scalar quantity of something situated at a certain point p f , the potential of u' for any point p is a scalar function of p, defined by the equation pot? A u [p'-p]< and the Hewtonian of u' for any point p is a vector function of p defined by the equation new u'—Y^i —Ws u '- [p — P]o Again, if ay is the vector representing the quantity and direction of something situated at the point //, the potential and the Laplacian of at' for any point p are vector functions of p defined by the equations poW= [^>y VECTOR ANALYSIS. 31 92. If u or (o is a scalar or vector function of position in space, we may write Potw, KewM, Potw, Lapo for the vol¬ ume-integrals of pot u', etc., taken as functions of <>'; i. e. we may set Pot u=fff pot y! dv 1 =fff~—^dv>\ New u=ff/Tie w u’ dv'v! dv', Pot oo^fff pot go' dv’=Jff^j~^dv', Lap ai=fff lap go' dv'=f/f ^^- x go' dv ', where the p is to he regarded as constant in the integration. This extends over all space, or wherever the u' or co' have any values other than zero. These integrals may themselves be called (integral) potentials, Newtonians, and Laplaeians. 93. <^PotM_ T) jdu ~7hT~ 0t (kd d P ot go doo dx ~ °^dx ‘ This will he evident with respect both to scalar and to vector functions, if we suppose that when we differentiate the poten¬ tial with respect to x, (thus varying the position of the point for which the potential is taken) each element of volume dv' in the implied integral remains fixed, not in absolute position , but in position relative to the point for which the potential is taken. This supposition is evidently allowable whenever the integration indicated by the symbol Pot tends to a definite limit when the limits of integration are indefinitely extended. Since we may substitute y and & for x in the preceding formula, and since a constant factor of any kind may be intro¬ duced under the sign of integration, we have p Pot w=Pot pu p.Pot G3 — Pot p.co /7 X Pot 03 —Pot pXGO p>p Pot u=Yot p.p$ LA^ p.p Pot 03 — Pot p.pGO i. e., the symbols p, p., pX, p.p may be applied indifferently before or after the sign Pot. Yet a certain restriction is to be observed. When the oper¬ ation of taking the (integral) potential does not give a definite finite value, the first members of these equations are to be regarded as entirely indeterminate, but the second members may have perfectly definite values. This would be the case for example, if u or co had a constant value throughout all 32 VECTOR ANALYSIS. space. It might seem harmless to set an indefinite expression equal to a definite, hut it would he dangerous, since we might with equal right set the indefinite expression equal to other definite expressions, and then be misled into supposing these definite expressions to be equal to one another. It will be safe to say that the above equations will hold, provided that the potential of u or to has a definite value. It will be observed that whenever Pot u or Pot to lias a definite value in general , (i. e. with the possible exception of certain points, lines, and surfaces),* the first members of all these equations will have definite values in general, and therefore the second members of the equations, being necessarily equal to the first members, when these have definite values, will also have definite values in general. 94. Again, whenever Pot u has a definite value, we may write u! V Pot u =Vfff-pdv'=ffJ'p- u' dv'. where r stands for [//—/>]„. But Avbence 1 P'-P 7 ~——w y g • iyo 17 Pot WzrrNeW U. Moreover, ISTew u will in general have a definite value, if Pot u has. 95. In like manner, whenever Pot to has a definite value, 7 X «*=V X ~ dv’=fff 7 y X of dv'. Substituting the value of 7 — given above we have 7 X Pot £e|f=Lap GO. Lap to will have a definite value in general, whenever Pot to has. 96. Hence, with the aid of No. 93, we obtain 7 X Lap go= Lap pX go, 7 . Lap go= 0. whenever Pot to has a definite value. 97. By the method of No. 93 we obtain 7 . N ewu=7. dv l —fff [7 u r . * Whenever it is said that a function of position in space has a definite value in general , this phrase is to he understood as explained above. The term definite is intended to exclude both indeterminate and infinite values. VECTOR ANALYSIS. 33 To find the value of this integral, we may regard the point p, which is constant in the integration, as the center of polar coordinates. Then r becomes the radius vector of the point p', and we may set dv'=r 2 clq dr, where r-dq is the element of a spherical surface having center at p and radius r. We may also set We thus obtain p' — p_du' r ~~ dr' p.New u=py~ dq dr=Anf-~dr=Ami' r==00 4 7tW r=() , where u denotes the average value of u in a spherical surface of radius r about the point y i( as center. Now if Pot u has in general a definite value, we must have u' — 0 for r= co . Also, p.New u will have in general a defin¬ ite value. For r— 0 , the value of u' is evidently u. We have, therefore, £7.New u——Anu, £7. £7 Pot U- — Ann. 98. If Pot co has in general a definite value, £7. £7 Pot 09= £7. £7|> i + Vj+W k] = p.yu % + £L pvj + £7. £7«> k, £ 7.£7 Pot 09=— Anoo. ITence, by No. 71, £7 X £7 X Pot GO — £7 £7.Pot GO — 4 7T GO. That is, Lap £7Xo5— New [/.Go—Anco. If we set 1 T — 1 AT go =^~ Lap i/X go, f» 2 =— New £ 7 . 03 , we have 03= G0 X + 03 2 , where co 1 and co. 2 are such functions of position that [ 7 .( 0 , = 0, and £7 Xc3 2 =0. This is expressed by saying that co l is solenoidcil , and op irrotational. Pot op and Pot op, like Pot co, will have in general definite values. It is worth while to notice that there is only one way in which a vector function of position in space having a definite potential can be thus divided into solenoidal and irrotational parts having definite potentials. For if o), + e, co 2 —e are two other such parts, \7 • 0 and £7 x £=0, 5 34 VECTOR ANALYSIS. Moreover, Pot e lias in general a definite value, and therefore 1 T 1 £=—Lap J7[|< .& — - — .New [ 7 . 8 = 0 . q. e. d. 4 7t 4 71 v 99. To assist the memory of the student, some of the princi¬ pal results of Nos. 93-98 may be expressed as follows : Let w 1 he any solenoidal vector function of position in space, o, any irrotational vector function, and u any scalar function, satisfying the conditions that their potentials have in general definite values. With respect to the solenoidal function cy, — Lap and pX are inverse operators ; i. e., 1 T 1 T — Lap ! 7 X 00=7 X— Lap Applied to the irrotational function <v 2 , either of these opera¬ tors gives zero ; i. e., Lap a) 2 =0, pX© 2 =0. With respect to the irrotational function ca 2 , or the scalar func¬ tion u, -U New and — p. are inverse operators ; i. e., 4 7t i X —— New [7.od= g% — 77 .— New u—u. '4 n 47T Applied to the solenoidal function co 1 the operator.p. gives zero; i. e., [ 7 . 00 = 0 , Since the most general form of a vector function having in general a definite potential may be written co, + a)„ the effect of these operators on such a function needs no especial mention. With respect to the solenoidal function a> l9 — Pot and pXpX are inverse operators; i. e., Pot (7X{7 X®!=/7X^ Pot j 7 X oo t —v X V X ~ Pot go 1 = co v With respect to the irrotational function cj 2 , ~ Pot and —pp. are inverse operators; i. e., Pot p y.oo g ^ -F^Potfziyg = - p p.— Pot a?* = . With respect to any scalar or vector function having in gen¬ eral a definite potential — Pot and -p.p are inverse opera¬ tors ; i. e., VECTOR ANALYSIS. 35 1 47T Pot [ 7 .p?u= — [7. J- Pot pu= — p.p—Pot U—U, 1 4 71 Pot p.p [dJ 1 + GJ 2 ]=—p.p 1 4 7T Pot [uq + — GJj + Ce7g. With respect to the solenoidal function w,, ~p ./7 and pXpX are equivalent: with respect to the irrotational function cq p.p and p p. are equivalent; i. e., j7.p 2 = 7p(y 2 100 . the interpretation of the preceding formulae .— Infinite values of the quantity which occurs in a volume-inte¬ gral as the coefficient of the element of volume will not neces¬ sarily make the value of the integral infinite, when they are confined to certain surfaces, lines, or points. Yet these sur¬ faces, lines, or points may contribute a certain finite amount to the value of the volume-integral, which must he separately calculated, and in the case of surfaces or lines is naturally expressed as a surface- or line-integral. Such cases are easily treated by substituting for the surface, line, or point, a very thin shell, or filament, or a solid very small in all dimensions, within which the function may be supposed to have a very large value. The only cases which we shall here consider in detail are those of surfaces at which the functions of position (u or co) are discontinuous, and the values of pu, p x w, p.co thus become infinite. Let the function u have the value u 1 on the side of the surface which we regard as the negative, and the value u 2 on the positive side. Let Au—u 2 — u 1 . If we substitute for the surface a shell of very small thickness a, within which the value of u varies uniformly as we pass through the shell, we shall have pu=v — within the shell a ’ v denoting a unit normal on the positive side of the surface. The elements of volume which compose the shell may be ex¬ pressed by a[daf where [>/u ] 0 is the magnitude of an element of the surface, clo being the vector element. Hence, pu dv = r Au \df 0 — Au da. Hence, when there are surfaces at which the values of u are discontinuous, the full value of Pot pu should always be under¬ stood as including the surface-integral /V ^ U ' 71 ' f/ UmprP relating to such surfaces. (Aid and da' are accented in the formula to indicate that they relate to the point f.) 36 VECTOR ANALYSIS. In the case of a vector function which is discontinuous at a surface, the expressions p.codv and pXtodv, relating to the element of the shell which we substitute for the surface of dis¬ continuity, are easily transformed by the principle that these expressions are the direct and skew surface-integrals of to for the element of the shell. (See Hos. 55, 56.) The part of the surface-integrals relating to the edge of the element may evi¬ dently he neglected, and we shall have j7 . go dv — co^.da— oo i .da= A co.da, j 7 X go dv=dax co^—dax Go^—dffxAoo. Whenever, therefore, to is discontinuous at surfaces, the expressions Pot p.to and Hew f .to must be regarded as implic¬ itly including the surface-integrals //r [p'-p] -A go'. da' and AA f p -p [p-p] 3 Ago' .da 1 0 respectively, relating to such surfaces, and the expressions Pot [?Xto and Lap pXto as including the surface-integrals and JT \x^k x[da ' xAw ' ] respectively, relating to such surfaces. 101. We have already seen that if to is the curl of any vec¬ tor function of position, p.to= 0. (Ho: 68.) The converse is evidently true, whenever the equation \ 7 .to —0 holds through¬ out all space, and to has in general a definite potential; for then go=!7 X — Lap go. K 47T 1 Again, if [7.to = 0 within any aperiphractic space A, contained within finite boundaries, we may suppose that space to be en¬ closed by a shell B having its inner surface coincident with the surface of A. We may imagine a function of position to', such that to' — to in A, to' = 0 outside of the shell B, and the integral fffto'.oo'dv for B has the least value consistent with the con¬ ditions that the normal component of to' at the outer surface is zero, and at the inner surface is equal to that of to. Then pr,co'—0 throughout all space, (Ho. 90,) and the potential of to' will have in general a definite value. Lienee, m = / 7 X^; Lap go , and to will have the same value within the space A. New Haven: Printed by Tuttle, Morehouse & Taylor, 1881. VECTOR ANALYSIS. 37 102. Def .—If co is a vector function of position in space, the Maxwellian * of co is a scalar function of position defined by the equation Max go — rrr/-, — • &)! ^ v '- ' [p'-p ] 3 0 (Compare No. 92.) From this definition the following prop¬ erties are easily derived. It is supposed that the functions co and u are such that their potentials have in general definite values. Max go — pr-. Pot go — Pot /7. go , p Max go — 1717 . Pot go — New [ 7 . go, Max [ 7 ti — —4 nu, An go = [7 x Fap go — [7 Max go. If the values of Lap Lap co, New Max co, and Max New u are in general definite, we may add 47r Pot go — Lap Lap go — New Max go, 4 7 t Pot u — — Max New u. In other words:—-The Maxwellian is the divergence of the potential, —and [7 are inverse operators for scalars and irrotatioual vectors, for vectors in general —— u Max is an An 1 operator which separates the irrotatioual from the solenoidal part. For scalars and irrotatioual vectors, 1 Max New and An ^ J New Max give the potential, for solenoidal vectors Lap Lap gives the potential, for vectors in general — New Max 47T gives the potential of the irrotatioual part, and ~~ Lap Lap the potential of the solenoidal part. 103. Def. —The following double volume-integrals are of frequent occurrence in physical problems. They are all scalar quantities, and none of them functions of position in space, as are the single volume-integrals which we have been consid¬ ering. The integrations extend over all space, or as far as the expression to he integrated has values other than zero. * The frequent occurrence of the integral in Maxwell’s Treatise on Electricity and Magnetism has suggested this name. 6 38 VICTOR ANALYSIS. The mutual potential , or potential product , of two scalar functions of position in space, is defined by the equation u w 1 Pot (u, w) — ffffff —ydv dv' = fffu Pot w dv = fff w Pot u dv. In the double volume-integral, r is the distance between the two elements of volume, and u relates to dv as w' to dv'. The mutual potential , or potential product , of two vector functions of position in space, is defined by the equation Pot (q>, CO) dv dv' — fff pNot go dv — fff oo .Pot cp dv. The mutual Laplacian , or Laplacian product , of two vector functions of position in space, is defined by the equation Lap (cp, go) = ffffff go dv dv' —fff 00 - P a P P dv ~ f/fP-^V 00 dv. The Newtonian product of a scalar and a vector function of position in space is defined by the equation New (u , go) = ffffff oo f 3 - u' dv dv’—fff ttf.New u dv. The Maxwellian product of a vector and a scalar function of position in space is defined by the equation Max (go, u) = ffffff u 9 J- -. go ' dv dv' — fff u M ax go dv = — New (u, go). It is of course supposed that u, w, cp, to are such functions of position that the above expressions have definite values. 104. By No. 97, 4:7ZU Pot W r= — £ 7 .New u Pot w — — p,[New u Pot ui\ -j- New u. New w. ■# The volume-integral of this equation gives 47 T Pot (u, w) = fff New u. New w dv, if the integral ffdff. N ew u Pot w VECTOR ANALYSIS. 39 for a closed surface, vanislies when the space included by the surface is indefinitely extended in all directions. This will he the case when everywhere outside of certain assignable limits the values of u and w are zero. Again, by No. 102, 47zr®.Pot cp = /7 xLap co • Pot (p —[7 Max ®.Pot cp — 17. [Lap go X Pot cp\ + Lap go . Lap cp — / 7 .[Max go Pot <p] + Max go Max cp. The volume-integral of this equation gives 47zr Pot (cp, go) — fff Lap cp . Lap® du + fff Max cp Max go dv, if the integrals ff dff . Lap go X Pot cp, ff do '. Pot cp Max go, for a closed surface, vanish when the space included by the surface is indefinitely extended in all directions. This will be the case if everywhere outside of certain assignable limits the values of cp and go are zero. 40 VECTOR ANALYSIS. CHAPTER III. CONCERNING LINEAR VECTOR FUNCTIONS. 105. Def .—A vector function of a vector is said to be linear , when the function of the sum of any two vectors is equal to the sum of the functions of the vectors. That is, if func.[p + p'] — func.[p] +func.[p'] for all values of p and //, the function is linear. In such cases it is easily shown that func. [ap + bp' -f op" + etc.] = a func. \p\ + b func.[p'] + c func. [p"] + etc. 106. An expression of the form a A.p + yd p.p-f etc. evidently represents a linear function of p, and may he con¬ veniently written in the form The expression or |«A + ySp 4-etc. }.p. p.aX + p./3 p-f-etc., p. {o'A + ftp + etc.}, also represents a linear function of p, which is, in general, different from the preceding, and will be called its conjugate . 107. Def .—An expression of the form al or ftp will be called a dyad. An expression consisting of any number of dyads united by the signs + or — will be called a dyadic bino¬ mial, trinomial , etc., as the case may be, or more briefly, a dyadic. The latter term will be used so as to include the case of a single dyad. When we desire to express a dyadic by a single letter, the Greek capitals will be used, except such as are like the Roman, and also A and 1\ The letter I will also be used to represent a certain dyadic, to be mentioned hereafter. Since any linear vector function may be expressed by means of a dyadic, (as we shall see more particularly hereafter, see Ho. 110,) the study of such functions, which is evidently of primary importance in the theory of vectors, may be reduced to that of dyadics. VECTOR ANALYSIS. 41 108. Def .—Any two dyadics 0 and W are equal, when 0 .p = W.p for all values of p, or, when p. 0 = p. W for all values of p, or, when 0.0.p — 0. W.p for all values of 0 and of p. The third condition is easily shown to he equivalent both to the first and to the second. The three conditions are therefore equivalent. It follows that 0=¥,if 0.p= W.p, or p,0—p.W , for three non-complanar values of p. 109. Def .—We shall call the vector 0.p the (direct) product of 0 and p, the vector p. 0 the (direct) product of p and 0, and the scalar a. 0.p the (direct) product of a, 0 , and p. In the combination 0.p , we shall say that 0 is used as a prefactor, in the combination p. 0, as a postfactor. 110. If r is any linear function of p, and for p — i, p=j, p~l\ the values of r are respectively a, /9, and y, we may set and also r ={ai + fij+yJc}.p, . r — p.{ ia +j/3 + ky \. ~T— - 2 - ^ ~ ~ ~ I Rf > Therefore, any linear function may be expressed by a dyadic as prefactor and also by a dyadic as postfactor. 111. Def .—We shall say that a dyadic is multiplied by a scalar, when one of the vectors of each of its component dyads is multiplied by that scalar. It is evidently immaterial to which vector of any dyad the scalar factor is applied. The product of the dyadic 0 and the scalar a may be written either a0 or 0 a. The minus sign before a dyadic reverses the signs of all its terms. 112. The sign + in a dyadic, or connecting dyadics, may be regarded as expressing addition, since the combination of dyads and dyadics with this sign is subject to the laws of asso¬ ciation and commutation. 113. The combination of vectors in a dyad is evidently dis¬ tributive. That is, - -f K-f>ks P r=. V+C.C V + W-J + V+k.k. — — [o'-!- (3 + etc.] [A -|-p + etc.] = pA + + /9A-f fip +etc. We may therefore regard the dyad as a kind of product of the two vectors of which it is formed. Since this kind of product is not commutative, we shall have occasion to distinguish the factors as antecedent and consequent. 114. Since any vector may be expressed as a sum of i,j, and h with scalar coefficients, every dyadic may be reduced to a sum of the nine dyads n, ij, ik, ji, J, jk, hi, kp kk. 12 VECTOR ANALYSIS. (f a Qr^yUti'J 4 *• with scalar coefficients. Two such sums cannot be equal according to the definitions of No. 108, unless their coefficients are equal each to each. Hence dyadics are equal only when their equality can he deduced from the principle that the operation of forming a dyad is a distributive one. On this account, we may regard the dyad as the most gen¬ eral formmf product of two vectors. We shall call it the'Tnde- terminate product. The complete determination of a single dyad involves six independent scalars, of a dyadic, nine. 115. It follows from the principles of the last paragraph that if 2a fi = 2kA, then 2axfi = 2 xX A, and 2a.fi = 2 k. A. In other words, the vector and the scalar obtained from a dyadic by insertion of the sign of skew or direct multiplication in each dyad are both independent of the particular form in which the dyadic is expressed. We shall write 0 x and 0 B to indicate the vector and the scalar thus obtained. \ 0 x — (]. 0.7c—7c. 0. j) i -f- (7c. 0. i — i. 0.7c) j + ( i. 0,j —j. 0. i ) 7c, 0 S = i. 0. i + j. 0. j + 7c, 0.7c, as is at once evident, if we suppose 0 to be expanded in terms of ii, ij , etc. 116. Def. —The {direct) product of two dyads (indicated by a dot) is the dyad formed of the first and last of the four fac¬ tors, multiplied by the direct product of the second and third. That is, \afi\.\ yd j- — a fi.y d — fi.y ad. The (direct) product of two dyadics is the sum of all the jiro- ducts formed by prefixing a term of the first dyadic to a term of the second. Since the direct product of one dyadic with another is a dyadic, it may be multiplied in the same way by a third, and so on indefinitely. This kind of multiplication is evidently associative, as well as distributive. The same is true of the direct product of a series of factors of which the first and the last are either dyadics or vectors, and the other factors are dyadics. Thus the values of the expressions a.0.G.W./3 , a.0.G, 0.0.W.fi , 0 .G.W will not be affected by any insertion of parentheses. But this VECTOR ANALYSIS. 43 kind of multiplication is not commutative, except in the case of the direct product of two vectors. 117. Def .—The expressions 0 Xp and pX 0 represent dyad- ics which we shall call the skew products of 0 and p. If 0 = a\ + ft p + e tc., these skew products are defined by the equations 0 X p — ol A X p 4- £> p X p + etc. , p x 0 = p X ol A + p X ft p + etc. * It is evident that ) p x 0 }. W = p X { 0 • W 1 , V.{§ xp} = \¥.0\xp, \pX 0}-a — px\_0.ot\, a.{ 0xp} = [a.0]xp, IPX 0} X&= PX { t 0Xa\. We may therefore write without ambiguity p X r ^. W, W. <*> X p, p X or, m ® X p, p X o'. (VEc) e ~Ve (oe) (YEc.) f - Ve(lF) V{yec)F- vtVcF This may be expressed a little more generally by saying that the associative principle enunciated in Ho. 116 may be ex¬ tended to cases in which the initial or final vectors are con¬ nected with the other factors by the sign of skew multiplication. Moreover, a . p x = [OL X p\ ■ <jtJ au d { I J X p- <x = 0. [pX o']. These expressions evidently represent vectors. Bo ripx wxp\. ( i >. FVEc = (VcE)Ft* c.(VEf) These expressions represent dyadics. The braces cannot be omitted without ambiguity. 118. Since all the antecedents or all the consequents in any dyadic may be expressed in parts of any three non-complanar vectors, and since the sum of any number of dyads having the same antecedent or the same consequent may be expressed by a single dyad, it follows that any dyadic may be expressed as the sum of three dyads, and so, that either the antecedents or the consequents shall be any desired non-complanar vectors, but only in one wav when either the antecedents or the conse- v 9J quents are thus given. In particular, the dyadic * aii + bij + oik T o ji + bjj + c'jk + a"ki+b"kj + o"kk, 44 VECTOR ANALYSIS. which may for brevity he written and to where is equal to ai + {3j +yk, where a — ai + a j + a"k, (3 = hi + b'j + b"k, y — ci+c'j+c"k, iX +jp + kv, A = a i + bj + ck pi = a' i + b''j + c’k , v = a"i + b"J+c"k. 119. By a similar process, the sum of three dyads may he reduced to the sum of two dyads, whenever either the antece¬ dents or the consequents are complanar, and only in such cases. To prove the latter point, let us suppose that in the dyadic ocX -f m y + y v neither the antecedents nor the consequents are complanar. The vector {o'A-f ji )i + y r |. p is a linear function of p which will he paralle] to a when p is perpendicular to p and v, which will he parallel to ft when p is perpendicular to v and A, and which will he parallel to y when p is perpendicular to A and a. Hence, the function may he given any value whatever by giving the proper value to p. This would evidently not he the case with tlie sum of two dyads. Hence, by Ho. 108, this dyadic cannot he equal to the sum of two dyads. 120. In like manner, the sum of two dyads may he reduced to a single dyad, if either the antecedents or the consequents are parallel, and only in such cases. A sum of three dyads cannot he reduced to a single dyad, unless either their antecedents or consequents are parallel, or both antecedents and consequents are (separately) complanar. In the first case the reduction can always he made, in the second, occasionally. 121. Def. —A dyadic which cannot he reduced to the sum of less than three dyads will he called complete. VECTOR ANALYSIS, 45 A dyadic which can be reduced to the sum of two dyads will be called planar. When the plane of the antecedents coincides with that of the consequents, the dyadic will be called unbplanar. These planes are invariable for a given dyadic, although the dyadic may be so expressed that either the two antecedents or the two consequents may have any desired values (which are not parallel) within their planes. A dyadic which can be reduced to a single dyad will be called linear. When the antecedent and consequent are paral¬ lel, it will be called unilinear. A dyadic is said to have the value zero, when all its terms vanish. 122. If we set Ur=:#.p ? T = p.ft, and give p all possible values, a and r will receive all possible values, if <l> is complete. The values of a and r will be con¬ fined each to a plane, if 0 is planar, which planes will coincide, if 0 is uniplanar. The values of a and r will be confined each to a line if 0 is linear, which lines will coincide, if 0 is uni¬ linear. 123. The products of complete dyadics are complete, of complete and planar dyadics are planar, of complete and linear dyadics are linear. The products of planar dyadics are planar, except that when the plane of the consequents of the first dyadic is perpendicular to the plane of the antecedents of the second dyadic, the prod¬ uct reduces to a linear dyadic. The products of linear dyadics are linear, except that when the consequent of the first is perpendicular to the antecedent of the second, the product reduces to zero. The products of planar and linear dyadics are linear, except when, the planar preceding, the plane of its consequents is per¬ pendicular to the antecedent of the linear, or, the linear pre¬ ceding, its consequent is perpendicular to the plane of the antecedents of the planar. In these cases the product is zero. All these cases are readily proved, if we set o'— ft.v.p, and consider the limits within which a varies, when we give p all possible values. The products Wxp and pX 0 are evidently planar dyadics. 124. Def .—A dyadic 0 is said to be an idemfactor, when ft.p— p for all values of p, or when p. 0> — p for all values of p. 46 VECTOR AXALYSIS, If either of these conditions holds true, 0 must he reducible to the form ii -\~jj + f'k. Therefore, both conditions will hold, if either do. All such dyadics are equal, by ho. 108. They will be represented by the letter I. The direct product of an idem factor with another dyadic is equal to that dyadic. That is, I. <b — <I J , ']>. I r= where 0 is any dyadic. A dyadic of the form aa' + (ip 1 + in which a', [i\ y' are the reciprocals of a, p, y, is an idemfactor. (See No. 38.) A dyadic trinomial cannot be an idemfactor, un¬ less its antecedents and consequents are reciprocals. 125. If one of the direct products of two dyadics is an idem- factor, the other is also. For, if 0. W=I, (7. <i>. W = (7 for all values of <r, and 0 is complete; a. <i j . W. ( i j — G. ( i> for all values of <x, therefore for all values of a.0, and there¬ fore W.0— I. _ Def .—In this case, either dyadic is called the reciprocal of the other. It is evident that an incomplete dyadic cannot have any (finite) reciprocal. Reciprocals of the same dyadic are equal. For if 0 and W are both reciprocals of 12, <b — <b.D. W = W. If two dyadics are reciprocals, the operators formed by using these dyadics as prefactors are inverse, also the operators formed by using them as postfactors. 126. The reciprocal of any complete dyadic aX + p jj. + y Y is AV+ ///?'+ v’y’, 'where a\ ft', y' are the reciprocals of «, /?, p, and X\ ji , v are the reciprocals of /, //, >. (See No. 38.) VECTOR ANALYSIS, 47 127. Def *—We shall write 0~ x for the reciprocal of any (complete) dyadic 0, also 0 2 for 0.0, etc., and 0)~ 2 , for 0~ 1 .0~ 1 , etc. It is evident that 0~ n is the reciprocal of 0 n . 128. In the reduction of equations, if we have <i>. W = v.,t), we may cancel the 0 (which is equivalent to multiplying by <P~ r ) if 0> is a complete dyadic, hut not otherwise. The case is the same with such equations as a — #. p, w. 0 = £1.0, p.0— a. 0. 4 To cancel an incomplete dyadic in such cases would lie analo¬ gous to cancelling a zero factor in algebra. 129. T)ef .—If in any dyadic we transpose the factors in each term, the dvadic thus formed is said to he conjugate to the first. Thus u A + (1 p + yv an d A a -f- pfi + vy are conjugate to each other. A dyadic of which the value is not altered by such transposition is said to he self-conjugate. The conjugate of any dyadic 0 may he written 0 C 1 It is evi¬ dent that p.a>—(i) c .p and r i\p — p.'/y. 0 c .p and 0 .p are conjugate functions of p. (See Ho. 106). Since { 0 C } 2 — j 0 2 \c, we may write etc,, without ambi- guity. 130. The reciprocal of the product of any number of dyadics is equal to the product of their reciprocals taken in inverse order. Thus f\W.£lf'= £l-KW-£*-K The conjugate of the product of any number of dyadics is equal to the product of their conjugates taken in inverse order. Thus j *.w.n) c = n c ,w c .* c . Hence, since and we may write 0 ^ 1 without ambiguity. 131. It is sometimes convenient to he able to express by a dyadic taken in direct multiplication the same operation which would be effected by a given vector ( a ) in skew multiplication. The dyadic I X« will answer this purpose. For, by Ho. 117, 18 VECTOR ANALYSIS. {I Xa\.p =axp , p.|lx«'[ = p Xot, {IX a }. <P —a X c b. [I X a\ — X a. The same is true of the dyadic aX I, which is indeed identical with 1 X a, as appears from the equation I .{aX 1} = {IX a}.1. If a is a unit vector, |IX«} 2 = — {I— fttx], [IXo'| 8 = —I X a, {I X o'} 4 — I—cm, {I X o' | r> = IX a, etc. If /, j, h are a normal system of unit vectors, IX* = ixi = [j-jk, IX j = j X I = ik—hi , lXkz=zkxl— ji—ij • If a and ft are any vectors, [<*X/f] X I = I X [aX(3] — [3 a —a: [3. That is, the vector ax[3 as a pre- or post-factor in skew mul¬ tiplication is equivalent to the dyadic \fta—aft] taken as pre- or post-factor in direct multiplication. [a X ft] X p = | (3a—a[3 } .p, p X [a X /?] = p. | (3a— a(3 1. This is essentially the theorem of No. 2ft, expressed in a form more symmetrical, and more easily remembered. 132. The equation a (3 X y h (3 y X a + y a X (3 — a. (3 X y I gives, on mnltiplication by any vector p, the identical equation p.a (3xy + p-(3 y X a p.y a X [3 — a.(3 Xy p. (See No. 37.) The former equation is therefore identically true. (See No. 108.) It is a little more general than the equation aa A- [3(3' y yy' — I, which we have already considered (No. 124), since, in the form here given, it is not necessary that a, ft, and y should be non- complanar. We may also write VECTOR ANALYSIS. fiXy at + yxot fi + axfi y — oL.fi x y I. Multiplying this equation by p as prefactor, (or the first equa¬ tion by p as postfactor,) we obtain ft. fi X V a + ft. y X ol fi + ft.a x fi y — ol. fix y ft. (Compare No. 87.) For three complanar vljjftors we have ol fix y + fi y X ol + y a X fi = 0 . Multiplying this by v, a unit normal to the plane of «, fft and y, we have ol fi X y. y + fi yXoi. v + y aX fi. v = 0. This equation expresses the well-known theorem that if the geometrical sum of three vectors is zero, the magnitude of each vector is proportional to the sine of the angle between the other two. It also indicates the numerical coefficients by which one of three complanar vectors may be expressed in parts of the other two. 183. T)ef .—If two dyadics 0 and 0' are such that 0. ¥ = W.4>, they are said to be homologous. If any number of dyadics are homologous to one another, and any other dyadics are formed from them by the operations of taking multiples, sums, differences, powers, reciprocals, or products, such dyadics will be homologous to each other and to the original dyadics. This requires demonstration only in regard to reciprocals. Now if 0.W — W.&, W. 0- 1 == 4 * - 1 .0. W. 0 1 =Z0- 1 .W.0.0~ 1 = 0^ 1 . W. That is, is homologous to W, if 0 is. IB!. If we call ¥.0 _1 or 0~ 1 .W the quotient of W and <P, we may say that the rules of addition, subtraction, multiplica¬ tion and division of homologous dyadics are identical with those of arithmetic or ordinary algebra, except that limitations analogous to those respecting zero in algebra must be observed with respect to all incomplete dyadics. It follows that the algebraic and higher analysis of homol¬ ogous dyadics is substantially identical with that of scalars. 135. It is always possible to express a dyadic in three terms, so that both the antecedents and the consequents shall be per¬ pendicular among themselves. To show this for any dyadic 0, let us set ft 7 ft’~'T>.ft, 50 VECTOR ANA LYSIS. p being a unit-vector, and consider the different values of p' for all possible directions of p. Let tlie direction of the unit vector i be so determined that, when p coincides with b, tlie value of [>' shall be at least as great as‘for any other direction of p. And let the direction of the unit vector j be so deter¬ mined that when p coincides with j % the value of p' shall be at least as great as for any other direction of p which is perpen¬ dicular to b. Let k liave its usual position with respect to i and /'. It is evidently possible to express d> in the form We have therefore and o' i -)- fij -(- yk. p ~ J on -f- fij -f- yk j. ft, dft — j cci 4- fij -)- yk \ Aft. .Now the supposed property of the direction of i requires that when p coincides with b and dp is perpendicular to /, dp' shall be perpendicular to p', which will then be parallel to a. But if dp is parallel to j or k, it will be perpendicular to i, and dp' will be parallel to ft or y, as the case may be. Therefore ft and y are perpendicular to a. In the same way it may be shown that the condition relative to j requires that y shall be perpen¬ dicular to ft. IVe may therefore set f i> z=z ai'i -f bfj -f- ck'k, where bf jj kj like b, j, k, constitute a normal system of unit vectors (see No. 11), and a, b. c are scalars which may be either positive or negative. It makes an important difference whether the number of these scalars which are negative is even or odd. If two are negative, say a and b , we may make them positive by reversing the directions of i ! and j'. The vectors bj jj k! will still con¬ stitute a normal system. But if we should reverse the direc¬ tions of an odd number of these vectors, they would cease to constitute a normal system, and to be superposable upon the system i,j, k. We may, however, always set either or *T> — ai'i + bfj + ck’k, <T> —: — { ai'i -|- bfj 4- ck'k }, with positive values of a, b, and c. At the limit between these cases are the planar dyadics, in which one of the three terms vanishes, and the dyadic reduces to the form ai’i + bj'j, in which a and b may always be made positive by giving the proper directions to b' and j'. VECTOR ANALYSIS. 51 If the numerical values of a, b, c are all unequal, there will be only one way in which the value of 0 may be thus expressed. If they are not all unequal, there will be an infinite number of ways in which 0 may be thus expressed, in all of which the three scalar coefficients will have the same values with excep¬ tion of the changes of signs mentioned above. If the three values are numerically identical, we may give to either system of normal vectors an arbitrary position. 136. It follows that any self-con jugate dyadic may be ex¬ pressed in the form ail + bjj + ckk. where i, j, k are a normal system of unit vectors, and a, b, c are positive or negative scalars. 137. Any dyadic may be divided into two parts, of which one shall be self-conjugate, and the other of the form I X«. These parts are found by taking half the sum and half the difference of the dyadic and its conjugate. It is evident that <i> = £{<!>+ (I) c ! + 1 : rI) ~ % } • low 1 j 0 -f- (J\ : j is self-con jugate, and W 11 — <2> 0 } = Ix[--£* J|. (See No. 131.) Rotations and Strains. 138. To illustrate the use of dyadics as operators, let us sup¬ pose that a body receives such a displacement that P' = &.p, [> and p' being the position-vectors of the same point of the body in its initial and subsequent positions. The same relation will hold of the vectors which unite any two points of the body in. their initial and subsequent positions. For if <> v p 2 are the original position-vectors of the points, and />/, <p their final position-vectors, we have Pi —P 1? P'2 ~~ AP;j 5 whence P'2 Pi -^'[Ps PlJ* In the most general case, the body is said to receive a homo¬ geneous strain. In special cases, the displacement reduces to a rotation. Lines in the body initially straight and parallel will be straight and parallel after the displacement, and sur¬ faces initially plane and parallel will be plane and parallel after the displacement. 52 VECTOR ANALYSIS. 139. The vectors (a, a') which represent any plane surface in the body in its initial and final positions will be linear func¬ tions of each other. (This will appear, if we consider the four sides of a tetrahedron in the body.) To find the relation of the dyadics which express a' as a function of <7, and // as a function of p, let p — ■; a\ + ftp + Y v .}. p. Then, if we write //, 1 / for the reciprocals of /, p, v, the vectors //, F become by the strain «, /9, y. Therefore the surfaces //.'xV, vxX, A'Xp' become /9 Xy, yXa , a X fi. But P-'X'X, y'xX, A X [J. are ithe reciprocals of p X v, v x /, / X p. The relation sought is therefore < 3 ’ =4 ft x y M x y + y X ot v x A + a >&/i A X p }. cr. 140. The volume X'.p'Xv' becomes by the strain a.fixy. The unit of volume becomes therefore (a. ft x y) {A.pXv). —Tt follows that the scalar product of the three ante¬ cedents multiplied by the scalar product of the three conse¬ quents of a dyadic expressed as a trinomial is independent of the' particular form in which the dyadic is thus expressed. This quantity is the determinant of the coefficients of the nine terms of the form aii -f bij + etc., into which the dyadic may be expanded. We shall call it the determinant of the dyadic, and shall denote it by the notation S 1 / \<P\ when the dyadic is expressed by a single letter. If a dyadic is incomplete, its determinant is zero, and con¬ versely. iS The determinant of the product of any number of dyadics is equal to the product of their determinants. The determi¬ nant of the reciprocal of a dyadic is the reciprocal of the deter¬ minant of that dyadic. The determinants of a dyadic and its conjugate are equal. The relation of the surfaces o’Xo may be expressed by the equation a' — |<i> | ^c" 1 cr. 141. Let us now consider the different cases of rotation and strain as determined by the nature of the dyadic ( Jk If is reducible to the form i,j, /•, j\ k! being normal systems of unit vectors (see JNTo. 11), the body will suffer no change of form. For if VECTOR ANALYSIS. 53 p = xi+yj+zk, we shall have pi = xi' + yf 4 - zk'. Conversely, if the body suffers no change of form, the opera¬ ting dyadic is reducible to the above form. In snch cases, it appears from simple geometrical considerations that the dis¬ placement of the body may be produced by a rotation about a certain axis. A dyadic reducible to the form i'i+jj+h'k may therefore be called a versor. 142. The conjugate operator evidently produces the reverse rotation. A versor, therefore, is the reciprocal of its conjugate. Conversely, if a dyadic is the reciprocal of its conjugate, it is either a versor, or a versor multiplied by —1. For the dyadic may be expressed in the form ai + fij+yk. Its conjugate will be iaA-j/3 + ky. If these are reciprocals, we have {oti -|- fij -f yk } . { la +j ft + ky | = aa + ftp 4- yy — I. But this relation cannot subsist unless a, /?, y are reciprocals to themselves, i. e., unless they are mutually perpendicular unit- vectors. Therefore, they either are a normal system of unit- vectors, or will become such if their directions are reversed. Therefore, one of the dyadics al 4- fij 4 - yk an d — ai — fij — yk is a versor. The criterion of a versor may therefore be written : f J>. f l J c — I, and | ® | =1. For the last equation we may substitute i<C >0, or ]. It is evident that the resultant of successive finite rotations is obtained by multiplication of the versors. 143. If we take the axis of the rotation for the direction of i, i' will have the same direction, and the versor reduces to the form ii+j'j + k'k, in which i, j, k and •/, j', k! a,re normal systems of unit vectors. 8 5 § VECTOR AISTALYSIS. « We may set / = cos qj + Iu ^ k, k' — cos q k — sin qj , and the versor reduces to or li + cos q\ jj+ kk\ -f sin q { hj —jit }, ii -f-cos q [I— ii } + sin q I x i, o*£ v<- «r-M rv\ cX*~r- where q is the angle of rotation, measured from j toward k, if the versor is used as a prefactor. 144. When any versor & is nsed as prefactor, the vector — 0x will be parallel to the axis of rotation, and equal in magnitude to twice the sine of the angle of rotation measured counter-clock-wise as seen from the direction in which the vector points. (This will appear if we suppose 0 to be repre¬ sented in the form given in the last paragraph.) The scalar 0 B will be equal to unity increased by twice the cosine of the same angle. Together, — 0 x and 0 B determine the versor without ambiguity. If we set 0 — the magnitude of 0 will be — 1 -f (? s 5 2 sin q 2 + 2 cos<? or tan \ q, t where q is measured counter-clock-wise as seen from the direc¬ tion in which 0 points. This vector d, which we may call the vector semitangent of version , determines the versor without ambiguity. 145. The versor 0 may be expressed in terms of 6 in various ways. Since 0 (as prefactor) changes a—dXa into a+Oxa (a being any vector), we have 00 + jl + lx<9'f s _ (1—0.0)1 + 200+21x0 1+0.0 1+676 : as will be evident on considering separately in the expression 0.p the components perpendicular and parallel to 0, or on sub¬ stituting in ii + cos q ( jj + kit) + sin q ( kj—jk ) for cos q and sin q their values in terms of tan \ q. If we set, in either of these equations, 0 = ai+bj+ck, VECTOR ANALYSIS. 55 we obtain, on reduction, tbe formula, (1 + u 3 — b 2 — c 2 )ii + (2ab—2c)ij+ (2ac + 2b)ik -f (2ab-\-2c)ji+ (1— a 2 + b 2 —c 2 )jj + (2bc—2a)]k + {2ac—2b)M+{2bc + 2a)kj + ( \—a 2 — b 3 + c 2 )kk 1+u 3 +5 3 +c 3 in which the versor is expressed in terms of the rectangular components of the vector semitangent of version. 146. If a, ft, y are unit vectors, expressions of the form 2aa—\ 2(3(3 —I, 2yy —I, are biquadrantal versors. A product like [2(3(3—\). [2 a a—1] is a versor of which the axis is perpendicular to a and /9, and the amount of rotation twice that which would carry a to /9. It is evident that any versor may be thus expressed, and that either a or j3 may be given any direction perpendicular to the axis of rotation. If <t> z=z[ 2(3(3—l}.[2aa— I}, and *P = [2yy—I}.[2/3fi— If, we have for the resultant of the successive rotations = { 2yy—I].[2aa—I]-. This may be applied to the composition of any two successive rotations, /? being taken perpendicular to the two axes of rotation, and affords the means of determining the resultant rotation by construction on the surface of a sphere. It also furnishes a simple method of finding the relations of the vector semitangents of version for the versors 0, I) and ¥. 0. Let *i = Then, since -0 X - 0 x _J !p ® l V _A Q — _ _2 Q _ l * • Z± * l-f'/y 3 i + yy 3 0 = 4 a.(3 (3a — 2 aa — 2(3(3 + I, f! __ a X(3 1 a.(3 ’ which is moreover geometrically evident. In like manner, Therefore, fiXy a.(i ’ S _ a Xy 3 ~ a.y • 6 x u - [ g Xfl X [fiXy] __ ax(3.y (3 1 3 a. (3 (3.y a. (3 (3.y — P' a P • X Y + P’P V X a + ( 3.y ax (3 a. (3 (3.y 56 VECTOR ANALYSIS. (See Ho. 38.) Tliat is, Also, a. ft (3.y‘ a.y Hence A — A — ( 1_ A’A)A + ®±-> ft _ A + A + A X A ' Q ^ - 1-^.0 which is the formula for the composition of successive finite rotations by means of their vector semitangents of version. 147. The versors just described constitute a particular class under the more general form a a’ -j- cos q{ fifi' + yy'} + sin q\ y/3 ! —j3y'}, in which «, /?, y are any non-complanar vectors, and /T, y' their reciprocals. A dyadic of this form as a prefactor does not affect any vector parallel to a. Its effect on a vector in the ft-y plane will be best understood if we imagine an ellipse to be described of which /3 and y are conjugate semi-diameters. If the vector to be operated on he a radius of this ellipse, we may evidently regard the ellipse with ;3, y, and the other vector, as the projections of a circle with two perpendicular radii and one other radius. A little consideration will show that if the third radius of the circle is advanced an angle q, its projection in the ellipse will be advanced as required by the dyadic prefactor. The effect, therefore, of such a prefactor on a vector in the (3-y plane may be obtained as follows :—Describe an ellipse of which /? and y are conjugate semi-diameters. Then describe a similar and similarly placed ellipse of which the vector to he operated on is a radius. The effect of the operator is to advance the radius in this ellipse, in the angular direction from ft toward y, over a segment which is to the total area of the ellipse as q is to 2 it. When used as a postfactor, the proper¬ ties of the dyadic are similar, hut the axis of no motion and the planes of rotation are in general different. Def .—Such dyadics we shall call cyclic. The Hth power (N being any whole number) of such a dyadic is obtained by multiplying q by H. If q is of the form 27rH/M (N and M being any whole numbers) the Mth power of the dyadic will be an idemfactor. A cyclic dyadic, therefore, may be regarded as a root of I, or at least capable of expression with any required degree of accuracy as a root of I. VECTOR ANALYSIS. It should be observed that tlie value of the above dyadic will not be altered by the substitution for a of any other parallel vector, or for /? and y of any other conjugate semi- diameters (which succeed one another in the same angular direction) of the same or any similar and similarly situated ellipse, with the changes which these substitutions require in the values of a', /?', y'. Or, to consider the same changes from another point of view, the value of the dyadic will not be altered by the substitution for a! of any other parallel vector or for [j and y' of any other conjugate semi-diameters (which succeed one another in the same angular direction) of the same or any similar and similarly situated ellipse, with the changes which these substitutions require in the values of a , /?, and f, defined as reciprocals of a', f, y'. 148. The strain represented by the equation p 1 = {aii -f- bjj 4- okJc }. p where a, b, c are positive scalars, may be described as consisting of three elongations (or contractions) parallel to the axes i, j, k, which are called the principal axes of the strain , and which have the property that their directions are not affected by the strain. The scalars a, b, c are called the principal ratios of elongation. (When one of these is less than unity, it repre¬ sents a contraction.) The ordgr of the three elongations is immaterial, since the original dyadic is equal to the product of the three dyadics aii-j-jj 4- kk, ii-\-bjj + ii-j- jjfckk taken in any order. Def .—A dyadic which is reducible to this form we shall call a right tensor. The displacement represented by a right tensor is called & pure strain. A right tensor is evidently self¬ conjugate. 149. We have seen (No. 135) that every dyadic may be expressed in the form =b | cti'i 4- bj'j -f ck'k }, where a, b , c are positive scalars. This is equivalent to dr { ai'i' + bj'j'-\-ck'k '}. { i’ipj'j+k'k | and to d= { i'if-j'j + k'k |. {aii -f bjgfckh }. Hence every dyadic may be expressed as the product of a versor and a right tensor with the scalar factor ±1. The versor may precede or follow. It will be the same versor in either case, and the ratios of elongation will be the same ; but 58 VECTOR ANALYSIS. the position of the principal axes of the tensor will differ in the two cases, either system being derived from the other by multiplication by the versor. Def .—The displacement represented by the equation P f = ~P is called inversion. The most general case of a homogeneous strain may therefore be produced by a pure strain ancl a rota¬ tion with or without inversion. 150. If (p — ai'i -f bj'j -j- ck'k, + b 2 j'j' + cattle', and f P c . <I> — a 2 ii -(- b 2 jj -)- c 2 kk. The general problem of the determination of the principal ratios and axes of strain for a given dyadic may thus be reduced to the case of a right tensor. 151. Def .—The effect of a prefactor of the form aaa' bftft - J- cyy 1 where a, b, c are positive or negative scalars, a, ft y non-com- planar vectors, and o!, /J 7 , y' their reciprocals, is to change a. into aa, ft into bft, and y into cy. As a postfactor, the same dyadic will change a! into cut!, ft' into bft, and y r into cy! Dvadics which can be reduced to this form we shall call tonic -* J _ (dr. reivio). The right tensor already described constitutes a particular case, distinguished by perpendicular axes and positive values of the coefficients a, b, c. The value of the dyadic is evidently not affected by sub¬ stituting vectors of different lengths but the same or opposite directions for a, ft y, with the necessary changes in the values of o!, ft, y ', defined as reciprocals of a, ft y. But, except this change, if a, b, c are unequal, the dyadic can be expressed only in one way in the above form. If, however, two of these coefficients are equal, say a and b, any two non-collinear vectors in the a-ft plane may be substituted for a and ft or, if the three coefficients are equal, any three non-complanar vectors may be substituted for a, ft y. 152. Tonics having the same axes (determined by the direc¬ tions of a , ft y) are homologous, and their multiplication is effected by multiplying their coefficients. Thus, { aa’+b t fij3'+c 1 ^}. { a 2 aa , -\-b^/30' + c 2 yy'} = { a % a 2 aa'-\- b ± b % e ± c 2 y y ' }. Hence, division of such dyadics is effected by division of their coefficients. A tonic of which the three coefficients a, b, c are / VECTOR ANALYSIS. 59 unequal, is homologous only with such dyadics as can he obtained by varying the coefficients. 153. The effect of a prefactor of the form aaa' + b{/3fd'-\-yy'\ + G {yfi'—/3y'}, or aaa' ~j -p cos q{/3fi'-\-yy'} -f -p sin q{yfi'—fiy'}, where a!, j3\ y' are the reciprocals of a, j3, y, and a, b, c, p, and q are scalars, of which p is positive, will be most evident if we resolve it into the factors aaa’ -j- (3(3' -j- yy r , a a + p/3 (3' -\-pyy', a a' U- cos q \ (3 (3'-\-yy' } + sin q { y ft—fly '}, of which the order is immaterial, and if we suppose the vector on which we operate to be resolved into two factors, one parallel to «, and the other in the {3-y plane. The effect of the first factor is to multiply by a the component parallel to a, without affecting the other. ’ The effect of the second is to multiply by p the component in the [3-y plane without affecting the other. The effect of the third is to give the component in the ftp plane the kind of elliptic rotation described in No. 147. The effect of the same dyadic as a postfactor is of the same nature. The value of the dyadic is not affected by the substitution for a of. another vector having the same direction, nor by the substitution for /? and y of two other conjugate semi-diameters of the same or a similar and similarly situated ellipse, and which follow one another in the same angular direction. Def. —Such dyadics we shall call cyclotonic . 154. Cyclotonics which are reducible to the same form except with respect to the values of a, p , and q are homolo¬ gous. They are multiplied by multiplying the values of a, and also those of p, and adding those of q. Thus, the product of a t aa' +p 1 cosq i {/?/?' + yy'\ + p 1 smq 1 {y/3'—fjy'} and a 2 aa' +p 2 cos q z + yy '\ + p 2 smq 2 {yf3'-f3y'} is a t a 2 aa'^-p 1 p^cos{q t + q 9 ) j (3(3' + yy' } +p 1 p^n(q i yq 2 ){y{3'-(3y'}. A dyadic of this form, in which the value of q is not zero, or the product of tt and a positive or negative integer, is homo¬ logous only with such dyadics as are obtained by varying the values of a, p , and q. 155. In general, any dyadic may be reduced to the form either of a tonic or of a cyclotonic. (The exceptions are such 60 VECTOR ANALYSIS. as are made by the limiting cases.) We may show this, and also indicate how the reduction may be made, as follows. Let 0 be any dyadic. We have first to show that there is at least one direction of p for which rp.p =. ap. This equation is equivalent to <P.p—ap — 0, or, \$—dL).p = 0. That is, (P=al is a planar dyadic, which may be expressed by the equation | 3 >— al\ — 0 . (See No. 140). Let (I> z=l A i + pj -f rk ; the equation becomes | [A— ai]i+[p—<xj\j + [v— ak~\k\ = 0, or, [A— ai] X [p—aj] . [v— ah] — 0, or, a 3 — (i.X+j.p +i.v)a i + (i.pxr+j.i'>m. + k.\xp)a — hxp.r = 0 . This may be written a* — r / J s a 3 + { } s | ® \a — ! | 0. Now if the dyadic (P is given in any form, the scalars * 8 , |®| are easily determined. We have therefore a cubic equation in a, for which we can find at least one and perhaps three roots. That is, we can find at least one value of a, and perhaps three, which will satisfy the equation |tf>—ai! = 0. By substitution of such a value, 0-al becomes a planar dyadic, the planes of which may be easily determined.* Let « be a vector normal to the plane of the consequents. Then { <1> —«I|. a-— 0, <I\a — aa. If 0 is a tonic, we may obtain three equations of this kind, say * in particular cases, $—al may reduce to a linear dyadic, or to zero. These, however, will present no difficulties to the student. VECTOR ANALYSIS. 61 <T>. a — aa, <P.ft = bft, V.y = cy, in which a, ft, y are not complanar. Hence, (by Ho. 108,) <l > = + bftft' + cyy', where a', ft', y' are the reciprocals of a, ft, y. In any case, we may suppose a to have the same sign as 101, since the cubic equation must have such a root. Let a (as before) he normal to the plane of the consequents of the planar (P—al, and a! normal to the plane of the antecedents, the lengths of a. and a! being such that a.a! — 1 .* Let ft he any vector normal to a!, and such that @.ft is not parallel to ft. (The case in which <P.ft is always parallel to ft, if ft is perpen¬ dicular to a', is evidently that of a tonic, and needs no farther discussion.) \<P—aI}.ft and therefore ®.ft will he perpendicu¬ lar to a! . The same will he true of @*.ft. How (by Ho. 140) [0.«].[0 3 ./5]X = ! 0| a.[&./3] X ft, that is, aa.[®*.ft>] X[ ( I J -ft] = l®l a.[$.ft’]xft. Hence, since [ f l j2 .ft]x[ ( f > .ft ] and [d>.ft]xft are parallel, ci[<i>*.ft] X [*-ft] =101 [*.ft] X ft. Since a~ 1 \@\ is positive, we may set p 2 = cr 1 |'H If we also set ft t =p-i<P.ft, ft 2 =p~*$ 2 .ft, etc., ft_ x —p<D- i.ft, ft _ 2 Jk^V-z.ft, etc., the vectors ft, ft t , ft 2 , etc., ft_ t , etc., will all lie in the plane perpendicular to a', and we shall have ft 2 Xft t =ft t Xft, [ft 2 + ft] X fti = o. We may therefore set ft % + ft = 2n ft t . Multiplying byy? - 1 0, and by ft 3 + ft 1 = 2 nft 2 , /i 4 + ft z = 2nft 3 , etc., ft 1 - 1 - ft„ t = 2 nft, ft + ft_ 2 = 2 nft_ 1 , etc. How, if n> 1 , and we lay off from a common origin the vectors ft > ftft etc., ft—t 5 ft— 2 ’ etc., * Eor the case in which the two planes are perpendicular to each other, see No. 157. 9 62 VECTOR ANALYSIS. the broken line joining the termini of these vectors will be convex toward the origin. All these vectors must therefore lie between two limiting lines, which may be drawn from the origin, and which may be described as having the directions of /Soo and /?_oo .* A vector having either of these directions is unaffected in direction by multiplication by 0. In this case, therefore, 0 is a tonic. If n<— 1, we may obtain the same result by considering the vectors ft i ft it ft %t ft 3’ ft 4 ) etG -, ft— i< ft—Z’ ft—3’ G *tC., except that a vector in the limiting directions will be reversed in direction by multiplication by 0, which implies that the two corresponding coefficients of the tonic are negative. If l,f we may set Then n = cos q. P- t + = 2 cos q /?. Let us now determine y by the equation ^ = cos q $ + sin q y. This gives (L-l = cos q§ — sin q y. How a! is one of the reciprocals of a, /?, and y. Let ft and y' be the others. If we set W =. (iosq{P@' + yy r } 4- sin q\yft' —fty'\, we have W.a = 0, WJ = = ft Therefore, since { aaa' +p W).a — aa — 0. n, {aaa' -\-pW}.@ =p@ 1 = *P.ft [aaa' +pW}.^_ t =p@ = it follows (by Ho. 108) that <P =z aaa' +pW = aaa' +p cos q{§§' + yy' } sin q{yft'—fty' }* 156. It will be sufficient to indicate (without demonstration) the forms of dyadics which belong to the particular cases which have been passed over in the preceding paragraph, so far as they present any notable peculiarities. * The termini of the vectors will in fact lie on a hyperbola, f For the limiting cases, in which n— 1, or n= — 1, see No. 156. VECTOR ANALYSIS. 63 If n= dtl, (page 62,) the dyadic may be reduced to the form aaa + b{ftft' + yy'} + hefty', where a, /3, y are three non-complanar vectors, a, (3', y 1 their reciprocals, and a, b, c positive or negative scalars. The effect of this as an operator, will be evident if we resolve it into the three homologous factors aaa' + ftft' + yy', aa' + b{ftft' + yy'\, aa 4- ftft' +yy' + cfty'. The displacement due to the last factor may be called a simple shear. It consists (when the dyadic is used as prefactor) of a motion parallel to /3, and proportioned to the distance from the a-j3 plane. This factor may be called a shearer. This dyadic is homologous with such as are obtained by vary¬ ing the values of a , b, c, and only with such, when the values of a and b are different, and that of c other than zero. 157. If the planar <P-al (page 61) has perpendicular planes, there may be another value of a, of the same sign as | @\, which will give a planar which has not perpendicular planes. When this is not the case, the dyadic may always be reduced to the form a{aa'-j-ftft' + yy'} + ab{aft'+ ft y'} + acay', where a, /3, y are three non-complanar vectors, a ', ft, y', their reciprocals, and a , b, e, positive or negative scalars. This may be resolved into the homologous factors al and I -1- 5] aft'+fty'} + cay’. The displacement due to the last factor may be called a complex shear. It consists (when the dyadic is used as prefactor) of a motion parallel to a which is proportional to the distance from the a-y plane, together with a motion parallel to b[3 +• ca which is proportional to the distance from the «-/3 plane. This factor may be called a complex shearer. This dyadic is homologous with such as are obtained by varying the values of a, b, c, and only such, unless b=0 . It is always possible to take three mutually perpendicular vectors for a , /?, and y; or, if it be preferred, to take such values for these vectors as shall make the term containing c vanish. 158. The dyadics described in the two last paragraphs may be called shearing dyadics. 64 VECTOR ANALYSIS. The criterion of a shearer is ) I } 3 = 0 . The criterion of a simple shearer is jtf-Ifs = 0. The criterion of a complex shearer is j<Z>-I} 3 =0, > 0. Note.—I f a dyadic 4> is a linear function of a vector p, (the term linear being used in the same sense as in No. 105,) we may represent the relation by an equation of the form $ = a/3 y.p + eC y.p + etc., or 4> = afiy + e£>/ + etc. y . p, where the expression in the braces may be called a triadic polynomial , and a single term a(3y a triad , or the indeterminate product of the three vectors a, /3, y. We are thus led successively to the consideration of higher orders of indeter¬ minate products of vectors, triads , tetrads , etc., in general polyads, and of polyno¬ mials consisting of such terms, triadics, teiradics, etc., in general polyadics. But the development of the subject in this direction lies beyond our present purpose. It may sometimes be convenient to use notations like P, v - h P, v h P,y P, rl to represent the conjugate dyadics which, the first as prefactor, and the second as postfactor, change a, /3, y into A, y, v, respectively. In the notations of the preceding chapter these would be written la' + pj3' + vy' and a'% + (i'p + y'v respectively, a', fi', y' denoting the reciprocals of a. (3, y. If r is a linear function of p, the dyadics which as prefactor and postfactor change p into t may be written respectively i —■ and —7. Ip pI If r is any function of p, the dyadics which as prefactor and postfactor change dp into dr may be written respectively dr \dp and dr In the notation of the following chapter the second of these, (when p denotes a posi¬ tion-vector), would be written yr. The triadic which as prefactor changes dp into cl) d 2 T d .—- may be written .—and that which as postfactor changes dp into —may be | dp I dp 2 dp | d 2 t written ——n. The latter would be written yyr in the notations of the following dp* | chapter. VECTOR ANALYSIS. 65 CHAPTER IY. [Supplementary to Chapter II.] CONCERNING THE DIFFERENTIAL AND INTEGRAL CALCULUS OF VECTORS. 159. If a) is a vector having continuously varying values in space, and p the vector determining the position of a point, we may set p — x i + yj + z A dp — dx i + dyj + dz />;, and regard to as a function of />, or of x , y, and v. Then, that is. If we set 7 , doo , g?go T t?co «g» — cm: — + ——(- dz —, tviK ■'$/ Ci'Z , , ( . c?g» Ago t Jgo ) dm = dp. j »-r- +.?-*- + * — <■ . cho Jco C«fcO Here /7 stands for G^GO = dp.\/GD. . d , d 7 d G?£C Gfe’ exactly as in Ho. 52, except that it is here applied to a vector and produces a dyadic, while in the former case it was applied to a scalar and produced a vector. The dyadic pto represents the nine differential coefficients of the three components of to with respect to x, y , and z, just as the vector yu (where u is a scalar function of p) represents the three differential coefficients of the scalar u with respect to x, y, and z. It is evident that the expressions p.to and yX to already defined (Ho. 54), are equivalent to jj7Goj s and \(7to\ x . 66 VECTOR ANALYSIS. 160. An important case is that in which the vector operated on is of the form pit,. We have then » dpu = dp.ppu, where d? u dx‘ ii + d 2 u .. d*u ... j -y _(_ _ _ J dxdy d^u dho .. rr*=< + 3 - Jx p + dy -33 + dxdz d 2 u . dydti ik ft )■ d*u , . d*u T . d 2 ii 7 7 -)——— —kb -f- — 7 —— kj -|- — kk. dzdx dzdy d?d J This dyadic, which is evidently self-con jugate, represents the six differential coefficients of the second order of u with respect to a?, y, and a* 161. The operators pX and p. may be applied to dyadics in a manner entirely analogous to their use with scalars. Thus we may define pX <P and p.(P by the equations Then, if Z t i Or, if px$ . d ( i j . dd> d ( J j FX lX Tx +jX ~dv + kX li (vih \AJ LJ \Aj& . d ( i j . d<T> 7 d$ r-'” = l -7f x +J -Ty + h ^- <I> — a i pj + y k, p-X®=pXai + p X P j + pXyk, p.V= p.ai +p.8 y + p.yk. <l>— ia +jB + k y, dy dp dy dz +j da dy dz dx + k dp dx da dp dy 1 — dx ^ dy dz' da dy 162 We may now regard p.p in expressions like p.pto as representing two successive operations, the result of which will be d 2 oi d 2 co d‘ 2 oi dx 2 dy* dz 2 in accordance with the definition of No. 70. We may also write p.p^ for * We might proceed to higher steps in differentiation by means of the triadics W«, vW w i the tetradics vw°! VVVV M t etc - See note on page 64. In like manner a dyadic function o£ position in space (<J>) might he differentiated by means of the triadic y<I>. the tetradic yy<t», etc. VECTOR ANALYSIS. 67 <r ® <p<i> d*® !h? + df + d?r although, in this case we cannot regard p.p as representing two successive operations until we have defined p0f That p.p(P=pp.<P—pXpX <P will he evident if we suppose 0 to he expressed in the form oh4- ftj-Pjh. (See hTo. 71.) 163. We have already seen that u"—vl —f dp.pu , where u' and u" denote the values of u at the beginning and the end of the line to which the integral relates. The same relation will hold for a vector %. e.. go" — go' =f dp. poo. 164. The following equations between surface-integrals for a closed surface and volume-integrals for the space enclosed seem worthy of mention. One or two have, already been given, and are here repeated for the sake of comparison. ffda u —fffdv pip (i ff da GD —fffdv pGD, ( 2 ) / ffda.GD -fffdv p.ao, 00 f ff da.® = fffdv p.®. (0 V /7 dax go = fffdv px go, ( 5 ) ffda X ® =fff dv px®. ( 6 ) X It may aid the memory of the student to observe that the transformation may be effected in each case by substituting fffdv? for //do,. 165. The following equations between line-integrals for a closed line and surface-integrals for any surface bounded by the line, may also be mentioned. (One of these has already been given. See hTo. 60.) f dpu — ff daxpu, ( 1 )' ~ fdp go =ff daxpoo, ( 2 ) fdp.GD=f/dcr.pxoj, (3)-"£<TF = fdp.&=ffda.px@, ( 4 ) *= j dp X go =ffp GD.dc> -ffdap. ca. ( 5 )^ y T/ _ _ ^ These transformations jnay be effected by substituting _ ^ ff\daxp] for ffdp. The brackets are here introduced to indicate that the multiplication of da with the 7, j, k implied in p is to be performed before any other multiplica- * See foot-note to No. 160. V ; _ ^Vrv.J ~ £ V .Vtyv.A/-JP — XI (vX — /V./Vv- 1 ?) X« ( 68 VECTOR ANALYSIS. tion which, may he required hy a subsequent sign. (This notation is not recommended for ordinary use, hut only sug¬ gested as a mnemonic artifice.) 166. To the equations in No. 65 may be added many others, as, /7[Mchj = f7U GJ + up GJ, (1) F[ix go] = [7TX GJ — 17&>XT, (2) |7X[lX©] = GJ.f/t — J7.r GJ — T. 17GJ + fr.GJT , (3) pr(r.Gj) = (7L GJ + [7CJ.T, (4) p'.{TGj}= 17.T GJ + T.£7GJ, (5) [7X {tgj}= [7XT GJ — TX{7GJ, (6) £7.{u$}— 1711 .$ + up.&, (7) etc. The principle in all these cases is that if we have one of the operators p, p., pX prefixed to a product of any kind, and we make any transformation of the expression which would he allowable if the p were a vector, (viz : hy changes in the order of the factors, in the signs of multiplication, in the parentheses written or implied, etc.,) hy which changes the p is brought into connection with one particular factor, the expression thus transformed will represent the paid of the value of the original expression which results from the variation of that factor. 161. From the relations indicated in the last four para¬ graphs, may he obtained directly a great number of trans¬ formations of definite integrals similar to those given in Nos. 14-71, and corresponding to those known in the scalar calculus hy the name of integration by parts. ' 168. The student will now find no difficulty in generalizing the integrations of differential equations given in Nos. 18-89 by applying to vectors those which relate to scalars, and to dyadics those which relate to vectors. 169. The propositions in No. 90 relating to minimum values of the volume-integral fff uco.codv may he generalized hy sub¬ stituting ( 0 . 0 .co for uco.a ;, 0 being a given dyadic function of position in space. 110. The theory of the integrals which have been called potentials, Newtonians, etc. (see Nos. 91-102) may be ex¬ tended to cases in which the operand is a vector instead of a scalar or a dyadic instead of a vector. So far as the demon¬ strations are concerned, the case of a vector may be reduced to that of a scalar by considering separately its three components, and the case of a dyadic may be reduced to that of a vector, by supposing the dyadic expressed in the form ipi+yj+cok and considering each of these terms separately. VECTOR ANALYSIS. 69 CHAPTER V. CONCERNING TRANSCENDENTAL FUNCTIONS OF DYADTCS. 171. Def -—The exponential f unction, tlie sine and the cosine of a dyadic may he defined by infinite series, exactly as the corresponding functions in scalar analysis, viz : These series are always convergent. For every value of 0 there is one and only one value of each of these functions. The exponential function may also be defined as the limit of the expression when 1ST, which is a whole number, is increased indefinitely. That this definition is equivalent to the preceding, will appear if the expression is expanded by the binomial theorem, which is evidently applicable in a case of this kind. These functions of 0 are homologous with 0. 172. We may define the logarithm as the function which is the inverse of the exponential, so that the equations e*= 0, W = log 0, are equivalent, leaving it undetermined, for the present, whether every dyadic has a logarithm, and whether a dyadic can have more than one. 173. It follows at once from the second definition of the exponential function that, if 0 and W are homologous, and that, if T is a positive or negative whole number, i f .t__ ae 10 TO VECTOR ANALYSIS. 174. If E and ¥ are homologous dyadics, and sncli that 0 = — £>, the definitions of Ho. 171 give immediately — cos <I> + E sin <I>, e~~-® z= cos 0 — E sin <I\ whence cos 0 = £.{, sin 0 =. — ^E{e A ®— 175. If 0.¥=¥.d>= 0, I 0+ W} 2 = 0 2 + ¥\ \ 0+Wy= 0 s + ¥\ etc. Therefore «* + *=«* + e*-I, cos | 0 + W }= COS 0 + cos W —I, sin | 0-\- W}= sin 0 + sin W. 176. For the first member of this equation is the limit of Ijl + N-i^H, that is, of |I + N-i$| N . If we set 0—aiE- ftj + jh, the limit becomes that of (l+N _1 o , .«4-N“ 1 ^.J + N _1 y.^) Isr , or (l + N -1 ^)^, the limit of which is the second member of the equation to be proved. 177. By the definition of exponentials, the expression e <i\ iq-jk y represents the limit of {I + q T$-i{kj-jk}}«. How I + ^H -1 \hj—'jk\ evidently represents a versor having the axis i and the infinitesimal angle of version pH -1 . Hence the above exponential represents a versor having the same axis and the angle of version q." If we set qi—co , the exponential may be written & I X 6) Such an expression therefore represents a versor. The axis and direction of rotation are determined bv the direction of to, and «/ 7 VECTOR ANALYSIS. 71 the angle of rotation is equal to the magnitude of co. The value of the versor will not be affected by Increasing or dimin¬ ishing the magnitude of oj by 2 tt. 178. If, as in Ho. 151, 0 = aaa' + bftft' 4- cyy', the definitions of Ho. 171 give e $ — e a aa ' _)_ e^ftft' + e c yy', cos 0 = cos a aa' -f cos b ft ft' + cos c yy', sin 0 — sin a aa! + sin b ft ft' + sin c yy'. If a, b, c are positive and unequal, we may add, by Ho. 172, log 0 — log a aa' + log b ft ft 1 + logc yy'. 179. If, as in Ho. 153, 0 — aaa' 4- b\ftft' + yy'\ + c{yft'—fty'} — aaa' + p cos q{ftft' + yy'} + p si nq{yft'—fty f }, we have by Ho. 173 V e®=e aaa \e h l ft ft'+ 77' 7ft'—ft7' 1 . But e <xaa' __ G tt aa ’ _J_ yy’ e b < PP'+yy' y=aa' + e^^ft' + yy'} \ yft'—fty' Y — aa ’ _j_ cos c {ftft' + yy'}-\- sin c {yft'—fty' [. Therefore, e® — e a aa' -f- e 1 * cosc{ftft'-{-yy'}-\- sin c\ yft' — fty'}. Hence, if a is positive, log 0 = log a aa' -f- iogp{ftft'+yy'}-\- q_\yft—fty'}- Since the value of 0 is not affected by increasing or dimin¬ ishing q by 2 tt, the function log 0 is many-valued. To find the value of cos 0 and sin 0, let us set © = b{ftft'-\-yy'}-yc{yft'—fty'}, 'B = yft' — fty'-' Then, by Ho. 175, cos 0 = cos { aaa'} -J- cos © — I. But cos j aaa'} — I = cos a a a' — aa', 72 VECTOR ANALYSIS. Therefore, cos ft = cos a a eft — a a + cos ©. Now, by No. 174, cos & = 4 {e~- e +c~ s - 0 }. Since £.© = e^- e = neb -f- e _c cosi{^'-|ly , } 4 - c~ c sin &{/(?'' — (fy'j, e~ ~' e — at at' + e c cos ?) ( 00 ' + 77 '}— e c sin 5 { 7 ^' — ^ 7 '}. Therefore cos©=u'n' , -j- 4 (e c +e~ c ) cos — £(<? c —e ~ c )sin b{yfi'—fiy'\, and cos 0 = cos a ota'-{-%(e c -\-e G ) cos — l(e c — e — c )$mb{y@ r —fty'} In like manner we find sin <P = sin a aa'-\-^(e c -\-e~ c ) sin -f-l(e c —e~ c ) cos b{y8'—@y'\. 180. If a, ft , y and eft, ft', y' are reciprocals, and ft = aaa' -|- b{@@'-\-yy'\ 4- cBy', and N is any whole number, < 2 > N = aftaoe 4 - b N \@@'-\-yy'} 4 - N b^^cBy'. Therefore, eft — e a a a' 4 - ft' {&§'ft-yy'}-{- ft'cfiy', cos ft — cos a a a' + cos b{fi@'-\-yy'} — csiu b fty\ sin ft — sin a a a' -j- sin b{§ft-ftyy'} + c cos b fty'. If a and b are unequal, and 0 other than zero, we may add log ft = log a a a' -f log b{§ft-\-yy'\ft- dr 1 ^. 181. If a , ft, y, and eft, ft', y' are reciprocals, and ft — al -f- b{a@'-\-py'\-{- cay', and N is a whole number, ft N = u N I + N« N -i b{ aft’ + ftft } + (N a N -i c + IN (N - 1 ) aft~^)ay'. Therefore VECTOR ANALYSIS. 73 e a l + e a b{aP'+Py'} + e a (ft*+c) ay', cos <P — cos al—b sin a{ a@' + @a’} — {^b 2 cos a-{-c sin a) ay', sin <P = sin a 1+5 cos a { aft + $ot! } — (^5 2 sin a—c cos a)ay'. Unless 5=0, we may add log f i> = log a I + bcr 1 { a@' + @a' \ -j- {car 1 —\b 2 ar 2 )ay'. 182. If we suppose any dyadic 0 to vary, but 'with the limitation that all its values are homologous, we may obtain from the definitions of Ho. 171 cl{e®\ = e ^. d*P, (>) d sfn <2> = cos CP . d<P , (2) d cos 0 z=i —sin 0 . d0, 0) dlog 0 =z (P^ 1 . d<i J , W as in the ordinary calculus, but we must not apply these equations to cases in which the values of 0 are not homologous. 183. If, however, F is any constant dyadic, the variations of tl will necessarily be homologous with tf, and we may write without other limitation than that T is constant, d sinj£U| dt = F. cos {tF}, d cos | tF } dt = - F sin {tF} d\og{tF } _I dt ~ t ‘ (i) (0 (0 (4) A second differentiation gives cP{e tv \ _ tT rr cP sin ;// ' ( _ dt 2 /A sini tl ) 3 d 2 cos{tF } dt 2 -F*.cos{tF}. (®) (6) (0 181. It follows that if we have a differential equation of the form dp dt ~ * P ’ the integral equation will be of the form 74 VECTOR ANALYSIS. p = e .p , ,tv ^ p' representing the value of p for t= 0. For this gives and the proper valne of p for t=0. 185. I)ef .—A flux which is a linear function of the position- vector is called a homogeneous-strain-flux from the nature of the strain which it produces. Such a flux may evidently be represented by a dyadic. In the equations of the last paragraph, we may suppose p to represent a position-vector, t the time, and P a liomogeneous- strain-flux. Then e tr will represent the strain produced by the flu x F in the time t. In like manner, if A represents a homogeneous strain, \ log A\/t will represent a homogeneous-strain-flux which would produce the strain A in the time t. 186. If we have where / is complete, the integral equation will he of the form p — e tl .a-\-e For this gives and a and /9 may be determined so as to satisfy the equations P(=o=“ + ft 187. The differential equation rl. 2 n will be satisfied bv t/ p = cos{tP}. a + sin {tr }. ft, FA — - r. sin j tP) . a + P. cos \tP }. <?, whence VECTOR ANALYSIS. 75 d 2 p dt 2 — F 2 . cos {tF }. a — F 2 . sin {tF \. (3 — —F 2 .p, \ bL v If /" is complete, the constants o. and ft may he determined to satisfy the equations P t -o O', \F\ = r.§. L dU t _ 0 188. If d 2 p dt 2 \F 2 —A 2 ), p, where r 2 —A 2 is a complete dyadic, and F.A = A.F= 0, we may set [i tr -tv tv p=\^e + %e + cos {£ y 4 } — — 1 which gives dp = U r J r _^F.e~ tV -tv fe +sin dt -A. sin{ t/1 } \. a ( tv —tv ) 4 -\\F.e +\I\e + A. cos{ tA } \. & dSp ={ir 2 .e T + ir*.e tT -A*.co8{tA\}.a dt"- ( fp _/p ) + \iF 2 .e -\F 2 .e ‘ -A 2 .sm{tA}\.8. - j F 2 -A 2 \. p. The constants a and ft are to be determined by (A — Of, r dp L dt — t =o = j F+Al.p. t- o 189. It will appear, on reference to hi os. 155-157, that every complete dyadic may be expressed in one of three forms, viz: as a sqnare, as a sqnare with the negative sign, or as a differ¬ ence of squares of two dyadics of which both the direct pro¬ ducts are equal to zero. It follows that every equation of the form , d 2 p df* = e - p where 0 is any constant and complete dyadic, may be inte¬ grated by the preceding formulae. 76 BIVECTOR ANALYSIS. NOTE cm BIVECTOK ANALYSIS.* 1. A vector is determined by three algebraic quantities. It often occurs that the solution of the equations by which these are to be determined gives imaginary values; i. e., instead of scalars we obtain biscalars, or expressions of the form a+eb, where a and b are scalars, and i—\/ — 1. It is most simple, and always allowable, to consider the vector as determined by its components parallel to a normal system of axes. In other words, a vector may be represented in the form ® i + yj + ^ Now if the vector is required to satisfy certain conditions, the solution of the equations which determine the values of x, y, and . 3 , in the most general case, will give results of the form X — X 1 -\- l x. 2 , y = y i + i y 2 = «i 4- where aq, rr 2 , y t , y 2 , values in we obtain z 1 , s 2 are scalars. * 1 + yj +« Substituting these (x t +ix 2 )i- f- {y y + iy z )j+ («,4 or, if we set * Thus far, in accordance with the purpose expressed in the foot-note on page 1, we have considered only real values of scalars and vectors. The object of this limitation has been to present the subject in the most elementary manner. The limitation is however often inconvenient, and does not allow the most symmetrical and complete development of the subject in many important directions. Thus in Chapter V, and the latter part of Chapter III, the exclusion of imaginary values has involved a considerable sacrifice of simplicity both in the enunciation of theorems and in their demonstration. The student will find an interesting and profitable exercise in working over this part of the subject with the aid of imaginary values, especially in the discussion of the imaginary roots of the cubic equation on page 60, and in the use of the formula e = cos $-M sin <P in developing the properties of the sines\ cosines, and exponentials of dyadics. BIVECTOR ANALYSIS. < ( we obtain p 1 =x 1 i + yj -f sq k, p 2 = x 2 i + yj + 2„ k, Pi 4 ~ L P%- We shall call this a bivector, a term which will include a vector as a particular case. When we wish to express a bivector by a single letter, we shall use the small German letters. Thus we may write * — Pi + 1 Ps- An important case is that in which y 1 and y 2 have the same direction. The bivector may then be expressed in the form (a+eb)p, in which the vector factor, if we choose, may be a unit vector. In this case, we may say that the bivector has a real direction . In fact, if we express the bivector in the form Oh+#2 ) i + k the ratios of the coefficients of i,j, and k, which determine the direction cosines of the vector, will in this case be real. 2. The consideration that operations upon bivectors may be regarded as operations upon their biscalar x- y- and .s-compo- nents is sufficient to show the possibility of a bivector analysis and to indicate what its rules must be. But this point of view does not afford the most simple conception of the operations which we have to perform upon bivectors. It is desirable that the definitions of the fundamental operations should be inde¬ pendent of such extraneous considerations as any system of axes. The various signs of our analysis, when applied to bivectors, may therefore be defined as follows; viz : The bivector equation y' ir' = fx" -f- iv" implies the two vector equations y! — y", and v' = r". — [y+iv] — —y -f z[— v]. [y' + iv'] + [y"-\-ir"] - [// + //] + i[v'~ fV|j [y' + iv'] . [y"+ir"] = [y'.y"~r'.r"] -f z [y'.v”+ r'.y"]. [y'+iv']x[y"+ LV "] = [p'Xy"-r T Xr"]+ i[y'xr"-\-r'xy"]. With these definitions, a great part of the laws of vector analysis may be applied at once to bi vector expressions. But an equation which is impossible in vector analysis may be pos¬ sible in. bivector analysis, and, in general, the number of roots 11 BIVECTOR ANALYSIS. hr Q i o of an equation, or of the values of a function, will he different according as we recognize, or do not recognize, imaginary values. 3. Def .—Two bivectors, or two biscalars, are said to be con¬ jugate, when their real parts are the same, and their imaginary parts differ in sign, and in sign only. Hence, the product of the conjugates of any number of bivectors and biscalars is the conjugate of the product of the bivectors and biscalars. This is true of any kind of product. The products of a vector and its conjugate are as follows: [M 4 iv ]. \ju— iv] = }a+}a -f y.y [fii+ ir] X \ fi~ir] — 2irXM [M+ lv \ [A *— ir ] — \ MM+ vv\ -f- i\ r/A—pir\. Hence, if /a and tv represent the real and imaginary parts of a bivector, the values of jA.ju+r.v, JuXu, /qn-j- w, y/x—jAV, are not affected by multiplying the bivector by a biscalar of the form a+tb, in which a 2 +b 2 = l. Thus, if we set we shall have f+ir 1 =z («-f ib)[jj- f- iv\ and That is, f.i' — iv' — [a — ib)\_fA — iv\ [q'+ — iv'\ = iv\. ja'.ja' 4- r'.y' = /u/I-4- y.y ; and so in the other cases. 4. Def .—In biscalar analysis, the product of a biscalar and its conjugate is a positive scalar. The positive square root of this scalar is called the modulus of the biscalar. In bivector analy¬ sis, the direct product of a bivector and its conjugate is, as seen above, a positive scalar. The positive square root of this scalar may be called the modulus of the bmector. When this modulus vanishes, the bivector vanishes, and only in this case. If the bivector is multiplied by a biscalar, its modulus is mul¬ tiplied by the modulus of the biscalar. The conjugate of a (real) vector is the vector itself, and the modulus of the vector is the same as its magnitude. 5. Def .—If between two vectors, a and f there subsists a relation of the form a = nS, where n is a scalar, we say that the vectors are parallel. BIYECTOE ANALYSIS. T9 Analogy leads us to call two bivectors parallel, when there subsists between them a relation of the form a = mb, where m (in the most general case) is a biscalar. To aid us in comprehending the geometrical signification of this relation, we may regard the biscalar as consisting of two factors, one of which is a positive scalar, (the modulus of the biscalar,) and the other may be put in the form cos q -f i sin q. The effect of multiplying a bivector by a positive scalar is obvious. To understand the effect of a multiplier of the form cos q -f t sin q upon a bivector p+tv, let us set p'+iv' = (cos q + i sin q)[p + iv\ We have then // = cos q p — sin q v , v f = cos q r + sin q p. How if p and v are of the same magnitude and at right angles, the effect of the multiplication is evidently to rotate these vectors in their plane an angular distance q, which is to be measured in the direction from v to p. In any case we may regard a and v as the projections (by parallel lines) of two per¬ pendicular vectors of the same length. The two last equations show that // and v will be the projections of the vectors obtained by the rotation of these perpendicular vectors in their plane through the angle q. Hence, if we construct an ellipse of which p and u are conjugate semi-diameters, p! and v' will be another pair of conjugate semi-diameters, and the sectors between p and p! and between u and i/, will each be to the whole area of the ellipse as q to 2 tt, the sector between v and v f lying on the same side of v as //, and that between p and p! lying on the same side of p as — v. It follows that any bi vector p+tv may be put in the form (cos q 4- i sin q) [n-f- z(5], in which a and J are at right angles, being the semi-axes of the ellipse of which p and v are conjugate semi-diameters. This ellipse we may call the directional ellipse of the bi vector. In the case of a real vector, or of a vector having a real direction, it reduces to a straight line. In any other case, the angular direction from the imaginary to the real part of the bivector is to be regarded as positive in the ellipse, and the specification of the ellipse must be considered incomplete without the indi¬ cation of this direction. Parallelism of bivectors, then, signifies the similarity and 80 BIVECTOR ANALYSIS. similar position of tlieir directional ellipses. Similar position includes identity of the angular directions mentioned above. 6. To reduce a given bivector r to the above form, Ave may set r.r = (cos q + i sin 1 ft].[a + ift] = (cos 2 q -f i sin 2 q) (a.a—ft.ft) — a+ ib Avliere a and b are scalars, Avhicli we may regard as known. The value of q may be determined by the equation b tan 2 q = a the quadrant to which 2 q belongs being determined so as to give sin 2 q and cos 2 q the same signs as b and a. Then a and ft will be given by the equation ift — (cos q — i sin q) r. The solution is indeterminate when the real and imaginary parts of the given biA r ector are perpendicular and equal in magnitude. In this case the directional ellipse is a circle, and the bivector may be called circular. The criterion of a circular bivector is r.r = 0. It is especially to be noticed that from this equation we can¬ not conclude that r = 0, as in the analysis of real vectors. This may also be shown by expressing r in the form xi+yj+zA', in Avhich x, y, z are biscalars. The equation then becomes x 2 -\-y 2 -\-z 2 — 0, which evidently does not require x, y, and s to vanish, as would be the case if only real values are considered. 7. Def. —We call two vectors p and a perpendicular when p,i r=0. Following the same analogy, Ave shall call two bisectors v and § perpendicular, when t.« = 0. In considering the geometrical signification of this equation, we shall first suppose that the real and imaginary components of r and § lie in the same plane, and that both r and § have not real directions. It is then evidently possible to express them in the form rn[a+ ift], ift'\, IT V ECTOR ANALYSIS. 81 Where m and m' are biscalar, a and $ are at right angles, and a! parallel with /?. Then the equation r.8=0 requires that D, and z=. 0. This shows that the directional ellipses of the two bivectors are similar and the angular direction from the real to the imag¬ inary component is the same in both, but the major axes of the ellipses are perpendicular. The case in which the directions of v and g are real, forms no exception to this rule. It will be observed that every circular bivector is perpen¬ dicular to itself, and to every parallel bivector. If two bivectors, y+tv, which do not lie in the same plane are perpendicular, we may resolve y. and v into components parallel and perpendicular to the plane of y! and v. The com¬ ponents perpendicular to the plane evidently contribute nothing to the value of [y+w ]. [p'-f iv'] Therefore the components of y and v parallel to the plane of //, i/, form a bi vector which is perpendicular to fj! -V tv. That is, if two bi vectors are perpendicular, the directional ellipse of either, projected upon the plane of the other and rotated through a quadrant in that plane, will be similar and similarly situated to the directional ellipse of the second. 8. A bivector may be divided in one and only one way into parts parallel and perpendicular to another, provided that the second is not circular. If a and 6 are the bivectors, the parts of a will be b.a U 6 aud a b.a b.b b. If b is circular, the resolution of a is impossible, unless it is perpendicular to b. In this case the resolution is indeterminate. 9. Since axb.a—-0, and aXb.b = 0, axb is perpendicular to a and b. We may regard the plane of the product as determined by the condition that the directional ellipses of the factors pro¬ jected upon it become similar and similarly situated. The directional ellipse of the product is similar to these projections, but its orientation is different by 90°. It may easily be shown that axb vanishes only with a or b, or when a and b are parallel. 10. The bivector equation (a X b. c) b — (b. c X b) a + (c. b X g) b — (b. a X b) c = 0 is identical, as may be verified by substituting expressions of the form xi+yj+zfc, (x, y, z being bi scalars,) for each of the bivectors. (Compare No. 37.) This equation shows that if the 82 BIVECTOR ANALYSIS. product ctXb of any two bivectors vanishes, one of these will be equal to the other with a biscalar coefficient, that is, they will be parallel, according to the definition given above. If the product ci.b X c of any three bivectors vanishes, the equation shows that one of these may be expressed as a sum of the other two with biscalar coefficients. In this case, we may say (from the analogy of the scalar analysis) that the three bivectors are complanar. (This does not imply that they lie in any same real plane.) If ct.bXc is not equal to zero, the equation shows that any fourth bivector may be expressed as a sum of a, b, and c with biscalar coefficients, and indicates how these coefficients may be determined. 11. The equation (r.ci) b X c + (v.b) c X ft + (r.c) ft X b = (ft X b.c) r is also identical, as may easily be verified. If we set and suppose that c = ci X b, r.ci=0, r.b = 0, the equation becomes (v.ftXb) axb = (aXb.aXb) r. This shows that if a bivector t is perpendicular to two bivectors a and b, which are not parallel, r will be parallel to ctXb. Therefore, all bivectors which are perpendicular to two given hi vectors, are parallel to each other, unless the given two are parallel. BIVECTOR ANALYSIS, 83 ADDENDA ET CORRIGENDA. Page 6, line 1 , for a=a + l3 read a=/3 + y. Page 8, No. 33, change signs of third member of equation. Page 11, line 7, after a.a'— (3.(M=y.y'=\. add as follows: a.f3'= 0, *a.y'= 0, (3.a'= 0, [3.y'=Q, y.a'— 0, y.(3'=0. These nine equations may be regarded as defining the relations between a, /?, y and afV, y' as reciprocals. Page 11, line 17, after y x a is add a.' x [V, ft' x y', y' x a', or Page 11, before No. 39, insert as follows: 38 a. If we multiply the identical equation (8) of No. 37 by a x r, we obtain the equation {a.(3 x y) ( p.(T x r)= a.p({3ya y. t—i3.t y.a) -4- j3.p(y.a a.T—y.r a.a) -f y.p(a.a [J.t — a.r (3. a ), which is therefore identical. But this equation cannot subsist identically, unless (a.j3 x y)a x T=a(j3.a y.T—j3.r y.a ) + (3(y.a a.T—y.r a. a) 4- y(a.a (3.r — a.r f3.o) is also an identical equation. (The reader will observe that in each of these equations the second member may be expressed as a determinant.) Prom these transformations, with those already given, it follows that a product formed of any number of letters (representing vectors and scalars), combined in any possible way by scalar, direct, and skew multiplications, may be reduced to a sum of products, containing each the sign x once and only once, when the original product contains it an odd number of times, or entirely free from the sign, when the original product contains it an even number of times. Page 15, line 7 from foot, in denominator of fraction, for dp dp ds ’ ds read d' 2 p d' 2 p ds' 2 ' ds' 2 ' Page 18, line 10 from foot, after continuous add and single-valued. Page 27, line 6, for u— constant read t—u= constant. Page 29, line 4, for ydu read y.Ow. Page 33, for second and third lines of No. 98, read V-V Pot« = v-V Pot \ui 3 -vj + ulc\ — v-Y Pot u % 4- v-V P°t V J + V-V Pot A* = — 4t xu % — 4 fcvj — 47TW Jc. = — 47ry. Page 36, line 5 from foot, after u, read aud that in the shell (compare No. 90). Page 36, line 4 from foot, dele (No. 90). A? ' ■ O . ' r " 3 *- * ftv L -4" ,1 » J& t' ' f - % ,i£z 1 r . ,.iT ; X.- v • v - «• - '•f 1 ./ t. , > . .. f r V- I'rr -h k 4 I .... >, ‘ ^ 1 .YvJf" ■ _ h \ *1 > «r< «* 4- * : __'^“ y MMM / ■ . , fc ... . „ ,, . f Jit, ,'rf/k #*0V1 • i l - ' i ■■■.; ; - V- ,?• » . ?.. J'u ■ * ' WI * 4dt 7 ' v - '■&.$' i {/- - J t, • ■ • '‘V „ cOKti :4 V. .> Y). r > - v 6 ' .. WC - : L \ 4 V -4 i + l^ »/ r i; & - VLv«r > <4 - I *■ I 1 '•/ /.i-f, / - V. < V f e$-jy - • - ... ; ■ ' i ! - 4 ' JL~ '.( -Oil • , r f ■ P " tpg 'mm 4 i. r '■ll '>4 m . •:' ■ ' 4 v . & it* . Y j ' " . |f '. V- . \ w$