Electromagnetic Theory, Vol. 1 (1893) — part 19 of 31

Theory of the Relative Motion of Electric Currents and the Medium.

§ 167. Thus, let us first abolish the intrinsic magnetisation of § 166, and substitute equivalent electric current. Here, by equivalent electric current, we mean a distribution of electric current which produces the same induction as the intrinsic magnetisation ; so that if both were to exist together, the induction would be everywhere doubled ; and then, if either of them were negatived, the resulting induction would be nil. That this is possible, whatever may be the distribution of inductivity, is a very remarkable property. The induction due to magnetisation is conditioned solely by the curl of the intrin- sic magnetic force, and the " poles " are essentially quite a secondary matter. That is, we may vary the poles as we like, provided we do not alter the curl of the intrinsic force, without affecting the induction or the associated energy.

The question then presents itself whether this equivalence continues to hold good when the medium is set in motion past the stationary magnetisation or electric current respectively. In the circuital equation (64), put h0 = 0, and introduce a term 00 on the right side, producing

curl(H-VDw) = C0 + cpE, . . . (75)

expressing the first circuital law ; whilst (65), or

curl(VwB-E) = /ijpE .... (76)

expresses the second. Now (75) only differs from (64) in the substitution of C0 here, for curlh0 there. That is, the new C0 and the old curlh0 are equivalent. We may, therefore,

282 ELECTROMAGNETIC THEORY. CH. III.

dismiss the idea of magnetisation, and let the source of the induction be any distribution of intrinsic electric current C0. With it is associated a certain distribution of induction, iden- tical with that due to any distribution of intrinsic magnetisation for which we have curlh0 = C0, and the equivalence persists when the medium is moving, or when the sources are moving. Corresponding to (70) we shall have

divB = 0, B = AF, curlF = C0/s, . (77) for the determination of the induction in the eolotropic manner previously pursued, when the steady state of affairs is reached j subject also to the previous reservation that w and curl B are perpendicular. That is to say, the induction is affected by the motion in the same way as if the medium had its inductivity decreased from /x to fts in the direction of motion, without any change transversely, whilst at the same time the source C0 is effectively increased to C0/s.

Equations (77) are suitable when the electric currents are in planes perpendicular to the direction of motion. For instance, let there be (to take a very easy example) two parallel plane sheets of electric current of surface-density C0 and - C0 respec- tively. The induction between them, which is B=//,C0 when at rest, becomes ftC0/s when they are in motion with velocity u perpendicular to their planes. Here s is the fraction (1 - u2/v*) as before, and the electric force accompanying -the changed in duction is simply the motional electric force.

There is another way of making the eolotropic comparison, namely, by employing the vector f instead of F. Then, instead of (77) we shall have

divB = 0, B = (A/«)f, curlf=C0. . (78) In this way of looking at the matter we regard 00 as suffering no change of effective strength, whilst the eolotropic operator X/s is such as to indicate increased inductivity (from p to /A/*) transverse to the motion, and unchanged inductivity parallel to it. The vector f is the excess of the magnetic force H of the flux over the motional magnetic force. Inasmuch as this way emphasises the fact that the intrinsic sources are really constant, it is not without its advantages. But the idea of increased transverse inductivity is rather an unmanageable one in general, especially in the case of moving electrification.

ELEMENTS OP VEOTORIAL ALGEBRA AND ANALYSIS. 283

Jf the sources and the medium have a common uniform translational motion, it may be seen from the general circuital equations that the steady distribution of the fluxes with respect to their sources is unaffected by the motion. That is, if we travel with the medium there is no change observable. This applies in the case of circuital sources (as curlh0 above), as well as divergent sources (as of electrification). The natural- ness of the result is obvious, when the relativity of motion is remembered.

The General Linear -Operator.

§ 168. It is now necessary to leave these special cases of eolotropy for fear of being carried away too far from the main subject, which is, the nature of linear vector operators, in ter- mination of this chapter on vector analysis. Up to the present only the symmetrical operator has been under consideration. Three rectangular axes are concerned, each of which is identi- fied with parallelism of the force and flux, or of a vector and a linear function thereof. Now, if we associate the algebraical ratio of the force to the flux with the three directions of parallelism, we see that the symmetrical linear operator depends upon three vectors. If they were arbitrary, this would involve nine scalar specifications ; but, being coperpendicular, there are really only six independent specifications ; and, moreover, if we transform to any other system of axes (independent or non-coplanar) there can still be no more than six independent data.

But it is easy to see that the general linear vector operator must involve nine scalar specifications, viz., three for each of the three independent axes of reference that may be chosen. Thus, let D be any linear function of E. We must then have, in terms of the i, j, k components, the following set of equa- tions : —

..... (79)

= C31E1 + C32E2 + C33E3>.

where the nine c's may have any values we please. The axes of reference are perpendicular only for convenience ; any set of three independent axes may be used, with nine properly deter-

284 ELECTROMAGNETIC THEORY. CH. IIL

mined c's to match. The relation between the vectors D and E, which is fully exhibited in (79), may be conveniently symbolised by

D = cE, ...... (80)

where c is the general linear operator.

If we exchange c12 and c21, &c., in the set (79) we shall usually change D, of course. Let it become D'. It is then fully given by the set

(81)

and these relations between D' and E may be symbolised, like (80), by the single equation

D' = c'E ....... (82)

The manipulation of sets of equations like (79) and (81) is lengthy and laborious. By the use of the linear operators, however, with proper attention to the laws governing them, the vectors may be manipulated with facility. Thus, to give the first and easiest example that presents itself, we may instantly turn AB to Ace^B. This is obvious enough, inas- much as the operation ii dicated by c"1 will be precisely can- celled by the operation c, so that cc-1B = B. But there is much more in it than that. For ActT^B is not merely the scalar pro- duct of the vectors A and cc1B, that is, of A and B, but is the scalar product of the vectors Ac and c1B, which are quite different. The vector Ac is the same as the vector c'A, whilst c-1B is the vector which, when operated upon by c, gives the vector B.

The constituents of the inverse operator c"1 may be found by solution of (79). Since D is a linear function of E, it follows that E is a linear function of D ; or the E's may be expressed in terms of the D's by means of a set of equations like (79), with new coefficients instead of the c's there. Similarly as regards the inverse operator c'"1 belonging to (81) and (82).

The vector formed by taking half the sum of the vectors D and D' is a symmetrical function of E, say

AK . . ... (83)

ELEMENTS OP VECTORIAL ALGEBRA AND ANALYSIS. 285

The constituents of X with two equal suffixes, namely, cn, C22» C33» are tne same as those of c and of c. But the other constituents are half the sum of the corresponding ones of c and c' ; for instance J(c12 + c21) in place of c12 and c21. From this we see that the operator A is its own conjugate, or is self- con jugate, making AE and A/E identical.

On the other hand, the vector formed by taking half the difference of D and D' is a simple vector product. By inspec- tion of (79) and (81), and remembering the structure of a vector product, we may see that

(84) where a is the vector given by

. (85)

We see, therefore, that any linear function, if it be not already of the symmetrical kind, may be represented as the sum of a symmetrical function and of a vector product. Thus, by addition and subtraction of (83) and (84) we obtain

cE = D = AE + VaE, ..... (86)

c'E = D' = AE-VaE ..... (87)

That is, in terms of the operators alone,

c = A + Va, ..... (88)

c' = A-Va, ..... (89)

where c (and therefore c') is general, whilst A is symmetrical.

The vector a is quite intrinsic, and independent of axes of refer- ence. The symmetrical operator A involves six scalars, the vector a three more, thus making up the full nine.

Notice, by (86), (87), that

.... (90)

or the scalar product of the force and the flux does not involve the vector a at all.

The Dyadical Structure of Linear Operators.

§ 169. Now go back to the equations (79). Observe that the right members, being the sums of three pairs of simple products,

286 ELECTROMAGNETIC THSORT CH. HI.

are themselves scalar products. Thus, let v&ree new vectors defined by

be introduced. Then we see that equations (79) are simply D^^E, D2 = c2E, D3 = c3E, . . (92)

and, therefore, by combining them, we produce the one vector equation

D = i.c1E+j.c2E + k.c3E, . . . (93)

thus proving that every linear operator may be exhibited in the dyadical form

c = i.c1+j.c2 + k.c3, .... (94)

whether it be symmetrical or not. The conjugate operator is, similarly,

c' = i.c'1+j.c'2 + k.c'3, .... (95)

where the accented vectors may be obtained from the un- accented given in (91) by changing c12 to cn, <fec., just as D' was got from D, in fact.

It is now easily to be proved that

cE = Ec', Ec = c'B, , . . (96)

that is, if we wish to remove an operator from before to behind a vector, we may do so by simply turning the operator to its conjugate ; that is, by putting on the accent, or by removing it if already there. It will, of course, be understood that when a, vector E follows c, and we use the dyadical form of c, as in (94), it is only the three second vectors that unite with E \ whereas, if E precedes c, it is the first vectors in c that unite with it. Thus :—

Ec = Ei.c1 + Ej.c2 + Ek.c3, . -. . (97)

c'E = i.c'1E+j.c'9E + k.c'3E. . . . (98)

These are identical vectors always. In the symmetrical case, the accents may be dropped, of course ; then EA, = A.E. It will be seen that the above way of writing and working dyadics fits in precisely with all the previous notation em- ployed in the vector algebra. The dots in (94) are merely

ELEMENTS OP VECTOBTAL ALGEBRA AND ANALYSIS. 287

separators, not signs of multiplication. When we put on a vector, before or behind, it unites with the vectors it is not separated from by dots ; we therefore obtain the forms (97), (98), in agreement with the principles of notation described in die early part of this chapter.

We have already pointed out that EAT, where A, is sym- metrical, may be regarded as the scalar product of E and AT, or of EA, (the same as AE), and T. The corresponding general property is the same ; that is, c being general, EcT is the scalar product of EC and F, or of E and cP. But EC is the same as c'E, and cP the same as Fc', so we have

EcF = c'EF = EFc' = Fc'E = FEc. . . (99)

The operator c must be united with one or other of the two vectors E and F, but it is immaterial which it is, provided we always change to the conjugate properly, as exemplified.

The same applies when there are many linear operators. Let there be three, a, 6, c. Then we have, to illustrate how the transformations are made,

. (100)

and so on. Here the brackets are introduced to show the association of the vectors and the operators.

We may also introduce anywhere a new operator, accom- panied, of course, by the reciprocal operator ; thus

cE[aF] = cB[db-lbf] = eE[a&-1(F&')] = &c. . (101)

The above will be sufficient to show the great power gained by the use of the operators, enabling one to do almost at sight algebraical work which would, in Cartesians, cover pages.

Hamilton's Theorem.

§ 170. The following little theorem, due to Hamilton, will serve to illustrate the working of vectors, as well as another purpose which will appear later.

Let m and n be a pair of vectors, then their vector product Vmn is perpendicular to both, by our definition of a vector pro- duct. That is,

0 = mVmn, 0 = nVmn, . . . (102)

288 ELECTROMAGNETIC THEORY. CH. III.

by our definition of a scalar product, and the parallelepipedal property. Now introduce cc~l betwen the V's and the vectors preceding them in (102). Thus

0 = mcc-1Vmn, 0 = ncc-1Vmn. . . (103)

These assert (and we cannot but believe it) that the vector c^Vmn is perpendicular to the vectors me and nc, or c'm and c'n ; it is, therefore, parallel to their vector product. The last statement is expressed by

«c-1Vmn = Vc/mc'n, .... (104)

where x is an unknown (or so far undetermined) scalar. Or, operating on both sides by c,

a;Vmn = cVc'mc'n. .... (105)

To find the value of x, we have merely to multiply (105) by a third vector, say 1. For this gives

zlVmn = IcVc'mc'n = c'lVc'mc'n, (106)

which gives the value of x explicitly, namely,

• • <107>

to be used in (105), which, with x thus settled, is the state- ment of the little (but important) theorem. The quantity x appears to depend upon 1, m, n. These vectors, however, enter into both the numerator and denominator in such a way that they can be wholly eliminated from x. This may be seen by expanding the numerator and denominator. In fact, a; is a pure constant, depending upon the operator c alone.

In the symmetrical case we may find its value by taking 1, m, n to be i, j, k, and the latter to be the principal axes. Thus

cl = ci = Cji, c'm = c«jj, c'n = c3k,

if clf c2, c3 are the principal scalar c's. So, by (107),

(108)

the continued product of the principal c's. In the general case we may put, as we know,

ELEMENTS OF VECTORTAL ALGEBRA AND ANALYSIS. 289

Using this in (107), on the understanding that clt c2, c3 are the principals of A now, we shall get, similarly,

.... (109)

which it will be a useful exercise to verify.

Since c is any linear operator, (105) remains true when for c we substitute its conjugate c', producing

aj'Vmn = c'Vcnicn, . . . . (110)

where x is got from the expression for x by changing c' to its conjugate c. But c and c only differ in the changed sign of a, as in (88), (89). On the other hand (109) contains a quadrati- cally, so that a reversal of its sign makes no difference. There- fore, x' is the same as x, which might not have been anticipated at the beginning of the evaluation.

Hamilton's Cubic and the Invariants concerned.

§ 171. The reader of Prof. Tait's profound treatise on Quaternions will probably stick at three places in particular, to say nothing of the numerous minor sticking-points that present themselves in all mathematical works of any value, and which may be readily overcome by the reader if he be a real student as well. First, there is the fundamental Chapter II., wherein the rules for the multiplication of vectors are made to depend upon the difficult mathematics of spherical conies, combined with versors, quaternions and metaphysics. Next, Chapter IV., where the reader may be puzzled to find out why the usual simple notion of differentials is departed from, although the departure is said to be obligatory. Thirdly, Chapter V., where the reader will be stopped nearly at the beginning by a rather formidable investigation of Hamilton's cubic. Only when the student is well acquainted with the nature of linear operators and how to work them can he tackle such an investigation. He should, therefore, pass on, to obtain the necessary experience. On then returning to the cubic he may find the investigation not so difficult after all, especially if it be simplified by some changes calculated to bring out the main points of the work more plainly.

The reader is led to think that the object of the investiga- tion is to invert a linear operator — that is, given D = cE, to find

u

290 ELECTROMAGNETIC THEORY. CIL III.

E = c~1D. But if this were all, it would be a remarkable example of how not to do it. For the inversion of a linear operator can be easily effected by other far simpler and more natural means. The mere inversion is nothing. It is the cubic equation itself that is the real goal. The process of reaching it is simplified by the omission of inverse operations. (It is also simplified by not introducing the auxiliary function called x-)

The fundamental cubic is derived from equation (105), or the equivalent equation (110) last investigated. I should remark that this equation is frequently useful in advanced vector-analysis as a transformation formula, turning Vcmcn to a function of Vmn. Now use the form (110), or

#Vmn = c'Vcmcn, ...... (HI)

where a; is a known function of c, viz.,

IVmn

Here c is any linear operator. Now if g be a constant, gm is a linear function of m, and therefore (c - g)m is any linear function of m; that is, c-g is a general linear operator. Equation (111) therefore remains true when we substitute c - g for c in it, not forgetting to make the change in x, and also the equivalent change in c', viz., from c' to c' — g. Equation (111) then becomes

xg Vmn = (c' - g)V(e - g)m(c - g)n, (113)

where xg is what x becomes by the change, that is, by (112),

x - (c-<7)lV(c

IVmn

Remembering that g is a constant, we may readily expand (113) to the form

= c'Vcmcn - g (Vcmcn + c Vmcn + c'Vcmn)

• #3Vmn +g'2 (c'Vmn + Vmcn + Vcmn), . . (115)

where the cubic function of g on the left side is the expansion of xg. The new coefficients xl and rc>2 are, of course, to be

ELEMENTS OF VECTORIAL ALGEBRA AND ANALYSIS. 291

found by expanding (114), but we do not want their values immediately. Since a; is a function of c only, xff is a function of c and g only, so that xl and X2 are functions of c only.

Now since (111) is an identity, so is (115) ; and since g may have any value, the last equation must be identically true for every power of g concerned, taken one at a time. That is, it splits into four identities. ~Now, on comparing the coefficients of g* on the left and right sides, we obtain (111) again. Simi- larly, we observe that the coefficients of g5 are the same. So far, then, we have nothing new. But the g and g2 terms give

. . (116) . . (117)

which are fresh identities. From these various others may be deduced by elimination or combination. The one we are seek- ing is obtained by operating on the first by c' and on the second by c'2, and then subtracting the second result from the first. This eliminates the vector in the brackets and leaves

• a?2c'2)Vmn = c'Vcmcn - c'3Vmn, . (118) = (a;-c'3)Vmn, . . . (119)

where the transition from (118) to (119) is made by usin& (111) again. Rearranging, we have the final result,

which is Hamilton's cubic. Note that Vmn may be any vector we please, and may, therefore, be denoted by a single letter. Observe, also, that the cubic function of c' is of the same form as that of g in the expansion of x.

Since (120) is true for all linear operators, we may write c instead of c', giving

x-xlC + x2c*-c* = 0 ..... (121)

This form is what we should have arrived at had we started from (105) instead of (110). The value of x is the same in either case, and it may be inferred from this that xl and x2 do not suffer any change when we pass from a linear operator to its conjugate.

That the complex function xg is independent of 1, m, n, may be seen by inspecting (114), and observing the parallel epipedal

u2

292 ELECTROMAGNETIC THEORY. <3H. III.

form of numerator and denominator. For, suppose we alter 1 to 1 + am, where a is any constant. The addition made to the numerator vanishes by the parallelepipedal property, and simi- larly in the denominator. The same invariance obtains when we change 1 to 1 + an. Also, the change of 1 to any scalar multiple of itself alters both the numerator and denominator in the same ratio. But, unless 1, m, n are coplanar, we may turn 1 to any vector by adding to it vectors parallel to 1, m and n of the right size. Therefore, xg is the same whatever vector 1 may be. By the same reasoning m may be any vector, and so may n. But 1, m, n should be independent vectors (that is, not coplanar). This is not because we can suppose that an actual discontinuity occurs when (for example) 1 is brought into the plane m, n, but merely because the expression x9 assumes an indeterminate form in the coplanar case.

From the invariance of xff follows that of the three functions it contains, namely, x, xv and xy That of #, of course, may be independently seen by itself, readily enough, but that of the others is less plain, because on expansion they are found to each involve three parallelepipedal products in the numerator instead of only one. Thus, expanding (114) we obtain

1 Vcmcn 4- m Vend + n Vclcm / 1 o o \

• -' ' ' (122)

cl Vmn -f cmVnl + en Vim / 1 o Q \

*2== lVmn~

In these we may, by the above, substitute c' for c, that is, turn the linear operator to its conjugate. This does not mean that xg is independent of the vector a which comes in when the operator is not symmetrical, but that it only involves a quadratically.

The value of x being

x = CjC^ + aca, . . . . . (124)

expressed in terms of the principal constants of the symmetrical operator and the rotation vector, we may obtain the reduced values of xl and X2 from it directly instead of from their general jxpressions. Thus : turn c to c-g (in 124), making it

$)* • (125)

ELEMENTS OF VECTORIAL ALGEBRA AND ANALYSIS. 293

On expansion, the coefficients of g and #2 show that

Xl = C2C3 + C3C1 + C1C2 + ft2> • • • (126) Z2 = Cj+C2 + C3.

Thus xz is independent of a, whilst #j involves its square, as we concluded previously.

When referred to the principal axes, Hamilton's cubic there- fore reduces to

0 = foc^ + aca) - (c^ + c2c3 + CgCj + a2)c

<';><} (127)

and in the symmetrical case of vanishing a to

Here the axes are coperpendicular. But we may disregard the principal axes altogether, and write the general cubic in the form

where the g'a are the roots of xg = 0. If, then, these roots are all real and different, there are three directions of parallelism of r and cr, or three r's such that

«i=0A» <**=9f9 <xz=yziy (130)

Now multiply the first by r2 and the second by TV giving

W-ftVi. ricr2=5'2r1r2 . . . (131)

The left members are not usually equal, but in the sym metrical case they are, and then the right members are equalised. Then 1^ = 0, or rx is perpendicular to r2. Simi- larly r2 is perpendicular to r3. This is the case of three mutually perpendicular axes of parallelism of force and flux we started from.

The Inversion of Linear Operators.

§ 172. If we write c~l for c in the cubic (121) we see that c-1r becomes expressed as a function of r, cr, and c2r. This is one way of inverting the operator, but a very clumsy way. The simple way is in terms of dyads, and is fully described by saying that if

c = a.l + b.m + c.n . ,. . . (132)

294 ELECTROMAGNETIC THEORY. CH. III.

is any linear operator in dyadical form, then its reciprocal is c-1 = L.A + M.B + N.C, . . . (133)

where A, B, C is the set complementary to a, b, c, and L, M, N the set complementary to 1, m, n. We have already used these complementary vectors in the early part of this chapter, equations (54) and (55), § 114. We showed by elementary considerations that any vector r could be expressed in terms of any three independent vectors a, b, c by

r = rA.a + rB.b + rC.c, . . . (134) where the complementary vectors A, B, C are got from a, b, c

by

A = I^, B = ^i, 0-™U (135)

aVbc aVbc aVbc v '

These equations serve to define the set complementary to a, b, c. We also showed at the same time that the vector r could be expressed in terms of the complementary set by

r = ra.A + rb.B + rc.O. .;-..' . (136)

To verify, we have merely to multiply (134) by A, B, C in turn, and (136) by a, b, c in turn.

These equations may be written

r = (a.A + b.B + c.C)r, . . . (137) r = (A.a + B.b + C.c)r, . ,, . (138)

showing that the dyadic in the brackets is of a very peculiar kind, inasmuch as its resultant effect on any vector is to repro- duce the vector. That is, taken as a whole, and disregarding its detailed functions, the dyadic is equivalent to unity. Now, suppose it is given that

R = (a.l + b.m + c.n)r, . . . (139) so that R is any linear function of r. We know that

R = (a.A + b.B + c.C)R, . . . (140)

and that r=(L.l + M.m + N.n)r, . . . (141)

by the property of the complementary vectors explained Comparing (140) with (139), we see that

lr = AR, mr = BR, nr = CR. . . (142)

ELEMENTS OP VECTORIAL ALGEBRA AND ANALYSIS. 295

Using these in. (141), we convert it to

r = (L.A + M.B + N.C)R, . . . (143)

comparing which with (139), we see that the inversion of the dyadic has been effected in a simple and neat manner.

Professor Gibbs calls the vectors A, B, 0 the reciprocals of a, b, c. They have some of the properties of reciprocals. Thus,

aA = l, bB = l, cO = l. . . . (144)

But it seems to me that the use of the word reciprocal in this manner is open to objection. It is in conflict with the obvious meaning of the recipocal a"1 of a vector a, that it is the vector whose tensor is the reciprocal of that of a with unchanged ort (or with ort reversed in the quaternionic system). It will also be observed that Gibbs's reciprocal of a vector depends not upon that vector alone, but upon two others as well. It would seem desirable, therefore, to choose some other name than re- ciprocal. I have provisionally used the word " complementary " in the above, to avoid confusion with the more natural use of "reciprocal."

Skew Product of a Vector and a Dyadic. The Differentia- tion of Linear Operators.

§173. In connection with dyadtcs, it should be remarked that we have only employed them in the manner they usually present themselves in physical mathematics, namely, so as to make direct products with the vectors they are associated with. But there is also the skew product to be considered in a com- plete treatment. Thus,

. . (145) . (U6)

Observe that, as before with the direct products, the vector r only combines with the vectors nearest to it in the dyadic, and not separated from it by the dots. Notice, too, that whereas the direct product <£r of the dyadic </> and a vector r is a vector ; on the other hand, the skew products V<£r and Vr<£ are themselves dyadics, and behave as such in union with vectors. Thus sV<£r is a vector, and so is sVr<£, the first being got by making scalar products of s with a, b, c,

296 ELECTROMAGNETIC THEORY. CH. Ill

and the second with Vra, &o. But to go further in this direction would be to go beyond the scope of the present treatment.

The differentiation of linear operators must, however, be mentioned, because the process is of frequent occurrence in electromagnetic investigations. If we think only of the differential coefficient of a scalar, that it is its rate of increase with some variable, it might seem at first sight that the differential coefficient of a linear operator was nonsense. But a little consideration will show that it is a perfectly natural, and by no means a difficult conception. Thus, from

D = cE, ...... (147)

where c is a linear operator, we obtain, by differentiation with respect to a scalar variable, say the time,

(148)

Here in the second term on the right we suppose that c is con- stant, and in the first that E is constant. The meaning of cE, then, is the rate of increase of D when c alone varies. It is the linear operator whose constituents are the rates of increase of the constituents of c. That this is so will be evident on differentiating the set of equations (79). Therefore, if

c=a.l + b.m + c.n, .... (149) we shall have

. . (150)

the sum of two dyadics. But it may be that our axes of reference are invariable, as for example when

c = i.c1+j.c2 + k.c3, .... (151)

i, j, k being a fixed set of rectangular orts. Then we have the one dyadic

.. (152)

where, of course, the dots over the i and j do not signify any differentiation.

In an isotropic dielectric of variable permittivity the electric stress leads to the force - Vc(^cE2) per unit volume, where the scalar c alone is differentiated, so that the result is - JE2Vc.

ELEMENTS OP VECTORIAL ALGEBliA AND ANALYSIS. 297

But if the dielectric be eolotropic, the corresponding force is This is equivalent to

• <153>

where dc/dx, &c., are differential coefficients of c as above ex- plained. It means the same as

-(VD-VE)(iED), .... (154)

where VD means that D alone, and VE that E alone is differentiated.

Summary of Method of Vector Analysis.

§ 174. In the last paragraph I came dangerously near to overstepping the imposed limits of my treatment of vectors, which is meant to present the subject merely in the form it issumes in ordinary physical mathematics. If we were to ignore the physical applications, and treat vector algebra as a branch of pure mathematics, regardless of practical limitations, there would be no bounds to the investigation of the subject. It will now be convenient to wind up with a few remarks on the previous, and on the nature and prospects of vector analysis.

Since we live in a world of vectors, an algebra or language of vectors is a positive necessity. At the commencement of this chapter, §§ 97 to 102, 1 made some general remarks on the nature of cartesian analysis, vector analysis, and quaternions ; and the reader is recommended to read them again from a more advanced point of view. Then, he was supposed to know next to nothing about vectors. Now, although he need not have absorbed all the special applications of vector algebra that have been given since, it may be presumed that he has acquired a general knowledge of the principles of the subject. There is no longer any question as to the desirability and utility of vectorial analysis. The present question is rather as to the form the vector algebra should take. On this point there is likely to be considerable difference of opinion, according to the point of view assumed, whether with regard to physical appli- cations, or abstract mathematical theory. Let us then, to begin with, summarise the leading points in the algebra given above.

First, there is the idea of the vector as a distinct entity. Nearly everyone nowadays knows and apprecktes the idea.

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And it is a noteworthy fact that ignorant men have long been in advance of the learned about vectors. Ignorant people, like Faraday, naturally think in vectors. They may know nothing jf their formal manipulation, but if they think about vectors, ihey think of them as vectors, that is, directed magnitudes. No ignorant man could or would think about the three com- ponents of a vector separately, and disconnected from one another. That is a device of learned mathematicians, to enable them to evade vectors. The device is often useful, especially for calculating purposes, but for general purposes of reasoning the manipulation of the scalar components instead of the vector itself is entirely wrong.

In order to facilitate the reading of vector work, the vector has its special type, thus E. This Clarendon, or any very similar neat black type (not block letters), is meant to mark the vector, always the vector, and never anything else, with the obvious exception of headlines that speak for themselves. Also, to economise letters, and ease the strain on the memory, the same letter suitably modified serves for the tensor, the ort, and the three scalar components. Thus E is the tensor, or size, and Ej the ort (signifying the orientation) so that E = EEX ; whilst the scalar components referred to rectangular axes are Ej, E2, E3. The physical dimensions may be most conveniently merged in the tensor E. Every vector has its species. The one we are most familiar with is the space vector, or straight line joining two points, or more strictly the displacement from one point to another. But all kinds of vector magnitudes are formally similar in having size and ort, and therefore, when considered vectorially, obey the same laws. The properties of the space vector therefore supply us with the rules or laws of vector algebra.

The addition and subtraction of vectors is embodied in the assertion (whose truth is obvious) that the sum of any number of vectors making (when put end to end) a circuit is zero ; or, in another form, all the various paths by which we may pass from one point to another are vectorially equivalent, namely, to the vector straight from point to point. Every vector equation therefore expresses this fact. Every term is a vector ; and if all the vectors be put on one side, with zero on the other, we express the circuital property ; whilst if we have vectors on both

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