Electromagnetic Theory, Vol. 1 (1893) — part 14 of 31

2VNH = 2VVH = 2curlH. . . . (162)

The interpretation may be more readily perceived by reversing the direction of the normal. Take N = - n, so that n is the normal drawn inward from the boundary. Then

2curlH + 2VnH = 0 ..... (163)

If H be magnetic force, curl H is the .electric current-density in the region. Now VnH is the surface equivalent of the bodily curlH. Ignore altogether the magnetic force outside the region, if there be any. Then the circuitation of H gives the current through a circuit. Applied to elementary circuits wholly within the region, the result is curl H. But at the boundary, where H suddenly ceases, there is a surface-current as well. To find its expression, apply the process of circuitation to a circuit consisting of two parallel lines of unit length, infinitely close together, but on opposite sides of the boundary, joined by infinitely short cross-pieces. Only the unit line inside the region contributes anything to the circuitation ; and by taking it to coincide with H, so as to make the circuitation a maximum, we find that VnH represents the surface-density of current. So, if J be current, we have, by (163), 2 J = 0 for any region, by itself. The surface-distribution and the volume-

ELEMENTS OP VECTORIAL ALGEBRA AND ANALYSIS. 197

distribution of current are complementary ; that is, they are properly joined together to make up a circuital distribution. Thus,

nVVH=-VVnH, .... (164)

where on the left side we have the divergence (at the surface) of the internal current, and on the right the equal convergence of the surface-current.

Five Examples of the Operation of V in Transforming from Circuital to Surface Summations.

§ 131. Next, take a few examples of the extended Theorem of Version (152), viz. : —

2F(T) = E/, .' . . . . (165)

where now, on the left side, we have the circuital summation of an odd function of T, and on the right an equivalent surface- summation, whose elementary part / is the value of F(T) for the circuit bounding the element of surface.

Taking three elements of surface to be unit squares, whose normals are i, j, k, we readily see that the corresponding /'s are

v2F(k) - v3F(j), v.FW-v^k), vjfl-vf®.. (166)

By means of these we can see the special form assumed by / in any case.

(a). Thus, take F(T) = T itself, the unit tangent. Then we have

2T = 0, ... . . . (167)

merely expressing the fundamental property of adding vectors, that the sum of any vectors forming, when put end to end, a circuit, is zero.

(6). Take F(T) = TP, where P is a scalar function of position. Then, with normal i, we have, by the first of (166),

; . . (168) so, writing N for i, we obtain

• .     .     (169) 
  

198 ELECTROMAGNETIC THEORY. OH. Til.

The quantity summed over the surface is, therefore, the surface representative of the curl of vP. This has no volume representative, its value being then zero.

(c). Take F(T) = TH. Here, with normal i to the element of surface, we have

/= V2kH - VgjH = V2H3 - V3H2 = iVVH, by (149). Therefore, putting N for i,

2TH = 2NVVH = 2NcurlH . . (170)

the Version Theorem again. But observe that, by the trans- formation (164), we may also write it

2TH = 2VNV.H, . . . . (171)

similarly to (169), in which the operand is a scalar. This is mnemonically useful, but (170) is more practically useful.

(d). Take F(T) = VTH. Here, with normal i, the first of (166) gives

/= V2VkH - V3VjH = V (kV2 - JV3)H.

But here we have kV2 - j V3 = ViV, so that /=V(ViV;H;

and therefore, generally, putting N for i,

2VTH = 2V(VNV)H. . . . (172)

(e). Let the quantity in the circuital summation be a vector of length P (a scalar function of position) drawn perpendi- cularly to the plane of T and N. That is,

F(T) = (VTN)P. We then find, taking N = i, and using the first of (166),

/= V2Vki . P - V3Vji . P = (j V2 + kV3)P = VP - i (iV) P. In general, therefore, 2 VTN . P = 2 (VP - N . NVP) = 2 VaP = 2 V (VNV)N.P . (173)

The element of the surface-summation is V8P, meaning the slope of P on the surface itself, disregarding any variation it

ELEMENTS OF VECTORIAL ALGEBRA AND ANALYSIS. 199

may have out of the surface. The last form of (173) involves

the transformation formula (52).

• Observe that in all the above examples,

2F(N) = 2F(V), (174)

when we pass from a closed surface to the enclosed region; and that

2F(T) = 2F(VNV), .... (175)

when we pass from a circuit to the surface it bounds. Thus, N becomes V, and T becomes VNv. But I cannot recommend anyone to be satisfied with such condensed symbolism alone. It is much more instructive to go more into detail, as in the above examples, and see how the transformations occur, bearing in mind the elementary reasoning upon which the passage from one kind of summation to another is based (§§ 128, 129).

Nine Examples of the Differentiating Effects of V»

§ 132. The following examples relate principally to the modi- fications introduced by the differentiating functions of V.

(a). We have, by the parallelepipedal property,

NVVE = VVEN = EVNV, . . . (176)

when V is a common vector. The equalities remain true when V is vex, provided we consistently employ the differentiating power in the three forms. Thus, the first form, expressing the N component of curl E, is not open to misconception. But in the second form, expressing the divergence of VEN, since N follows V, we must understand that N is supposed to remain constant. In the third form, again, the operand E precedes the differentiator. We must either, then, assume that y acts back- wards, or else, which is preferable, change the third form to VNV.E, the scalar product of VNv and E; or (VNv)E, if that be plainer.

(b). Suppose, however, that both vectors in the vector pro- duct are variable. Thus, required the divergence of VEH, expanded vectorially. We have

. . . (177}

200 ELECTROMAGNETIC THEORY. CH. IIL

where the first form alone is entirely unambiguous. But we may use either of the others, provided the differentiating power of y is made to act on both E and H. But if we keep to the plainer and more usual convention that the operand is to follow the operator, then the third form, in which E alone is differentiated, gives one part of the result, whilst the second form, or rather, its equivalent -EVvH, wherein H alone is dif- ferentiated, gives the rest. So we have, complete, and with- out ambiguity,

div VEH = H curl E-E curl H, . . . (178)

a very important transformation. It is concerned in the de- duction of the equation of activity from the two circuital laws of electromagnetism.

(c). In these circuital laws we have also to consider the curl of a vector product, viz., the curl of the motional electric force in one law, and the curl of the motional magnetic force in the other. Taking the former, we have

curlVqB = VVVqB, . ... (179)

where B is the induction and q the velocity of the medium supporting it. Apply the elementary transformation (52) to (179). It gives

VVVqB = q.VB-B.Vq, ... (180)

when y is a mere vector. But on the left side both q and B have to be differentiated ; therefore the same is true in both terms on the right side. This gives

VVVqB = qdivB + BV.q

-Bdivq-qV.B, . . . (181)

without ambiguity or need of reservation. That is to say, as in the q. vB of (180) both q and B have to be differentiated, we get qdivB when B alone, and Bv.q when q alone is differ- entiated. Similarly for the other term in (180).

Or we might write Vq when q alone, and VB when B alone suffers differentiation. Then, fully,

VVVqB = VVqVqB + VVBVqB, . . (182) VVqVqB = BV.q-Bdivq, . . (183)

WBVqB = qdivB-qV.B. . . (184)

ELEMENTS OP VECTORIAL ALGEBRA AND ANALYSIS. 201

Here the sum of (183) and (184) gives (181). The inean- iftg of By and qy has been already explained.

(d). Equation (181) may be applied to the circuital laws. Take the second, for example, in the form (4), § 66,

• curl (E - e0) = K + B + wo- - curlVqB, . (185)

and suppose that w = q, or that sources move with the medium. Then, by (181), we cancel the convective term wo-. Further,

we have B + qy.B = SB/fo, b7 (127)> §122» so that (185) becomes

• curl (E -e0) = K + SB/8* + Bdivq-BV.q, . (186)

and the corresponding form of the first law (equation (3), §66), is

curl(E-li0) = C + oD/o-* + Ddivq-Dy.q. . (187)

The time-variations refer to the same (moving) portion of the medium now. But if we wish to indicate the movement of electrification, &c., through the medium, that is, have relative motion u - q of p (and w - q of <r) with respect to the ether then to the right side of (187) add the term (u-q)/>, and to the right ride of (186) add (w - q)<r.

It is desirable to preserve the velocities u and w, or else the relative velocities, as well as the velocity q of the medium, in order to facilitate the construction and comprehension of problems relating to electromagnetic waves, which, although abstract and far removed from practice, are of a sufficiently simple nature to enable one to follow the course of events.

(e). We have already had the divergence of the product of a scalar and vector under consideration. Now examine its curl. Thus,

curlDP = VV(DP)

= VVD.P-VDV.P

= PcurlD-VDVP. . . . (188)

Here V can only make a vector product with D, because P is scalar. On the other hand, both P and D suffer differentiation. So in the second line we have V both before and after D.

(/). The divergence of the curl of any vector is zero. That is, divcurlH = 0, or VWH = 0. . - (189)

202 ELECTROMAGNETIC THEORY. CH. III.

If y here were a real vector (189) would mean that the volume of a parallelepiped vanishes when two edges coincide.

(g). A somewhat similar case is presented by the vanishing of the curl of a polar force. Thus,

curlVP = 0, or VV-VP = 0. . . . (190)

Of course Vw is zero. But the scalar product VV, or v2, is the Laplacean operator,

V2 = V12 + V22 + V32, .... (191) which occurs frequently.

(h). Let the operation curl be done twice on a vector. Thus,

(curl)2A = VVVVA

= V-VA-V2A, . . . (192) by the transforming formula (52). Or

V2A = VdivA-curl2A. . . . (193)

Thus there are two principal forms. If the vector A has no curl, then V2A is the slope of its divergence. If, on the other hand, it has no divergence, then - V2 has the same effect exactly as taking the curl twice.

(i). In the case of the operand being a scalar, then we have

V2P = divVP, (194)

the divergence of the slope of the scalar.

The Potential of a Scalar or Vector. The Characteristic Equation of a Potential, and its Solution.

§ 133. The last equation brings us to the theory of potentials. There are several senses in which the word potential has been employed, to enumerate which would be valueless here. For our present purpose we may conveniently fix its meaning by denning the potential at A of a quantity p at B to be the quantity />/47ir, where r is the distance from B to A. This is the rational potential, of course.

When p is distributed throughout space, whether at points, or over surfaces, or throughout volumes, the potential at any

ELEMENTS OP VECTORIAL ALGEBRA AND ANALYSIS. 203

pofnt- is the sum of the potentials of all the elements of p. That is,

, .... (195)

if P is the potential of p.

We may use the same definition when it is a vector that has to be potted, or potentialised. Thus, if A is the potential of 0, then

..... (196)

The summation is now a vector summation. Also, pot means " potential," or " the potential of," and has no more to do with kettle than the trigonometrical sin has to do with the un- mentionable one. It seems unnecessary to say so, but one cannot be too particular.

We may connect these potentials with v as follows : — Given that

divF = />, ..... (197)

that is to say, that the divergence of a vector F is p. The meaning of divergence has been explained more than once; both its intrinsic and its vectorial meaning. Now, if the vector F be explicitly given, it is clear that p is known definitely, since it is derived from F by differentiation, which should, perhaps, be regarded as a direct process, rather than inverse. But if it be p that is given, F is not immediately deter minable, unless we subject F to limitations. For we may construct any number of different F's to satisfy (197). Let every elementary source p of F send out the quantity p of F, according to my rational theory of sources already explained ; that is, the Unitarian system of one "line of force" to the unit "pole." Then, by the manner of construction, the resultant F will satisfy (197), and it will do so independently of the way we choose to let a source send out the flux it generates, whether equably or not.

But if it be done equally in all directions, so that p/4nr2 is the intensity of the " force " at distance r from the point-source p, and r1/o/47rr2 the vector force to correspond, where rx is a unit vector drawn from p towards the point under considera- tion, making the resultant F be

1, ..... (198)

204 ELECTROMAGNETIC THEORY. CH. IIL

we obtain a special solution of (197) which has a remarkable property, viz.,

curl F = 0, ..... (199;

so that if F be electric force, the voltage in any circuit is zero. The meaning of curl, I may observe, has been explained more than once ; both its intrinsic and its vectorial meaning. Those who seek can find, If they will not take the trouble to seek or to remember it is of no consequence to them. There are plenty of other things they may concern themselves about ; perhaps more profitably.

The property (199) is visibly true in the case of a single source. It is therefore separately true for the fields of all the sources, and therefore, by summation, is true for the com- plete F.

But (198) does not give the only vector which has no curl and a given divergence. For a constant vector (that is, constant throughout all space), has no curl and no divergence, unless we go to the very end of space to find the sources. Of course this constant solution has no relation to the sources p, and may be wholly ignored. If allowed, F would not vanish at an infinite distance from the sources. Remembering this, and excluding the constant solution, we may say that (198) is the solution of (197) and (199).

That there is no other solution may be proved analytically by Green's Theorem. But we do not really need any appeal to analysis of that kind, if the intrinsic meanings of divergence and curl are understood. For the admission that there could be a second solution, say, F + f, where F is the solu- tion (198;, would, by (197) and (199), imply that the vector f had no divergence, and also no curl anywhere. But the first of these conditions means that f is entirely circuital, if existent at all. The second denies that it is circuital. So f is non- existent.

Now observe that

or the slope of the scalar l/47rr is the vector with tensor l/4?rr2 and direction Tr It follows from this that

-Vpotp = F ...... (200)

ELEMENTS OP VECTORIAL ALGEBRA AND ANALYSIS. 205

when a single point-source is in question. Therefore, by sum- mation, the same is true for any distribution of sources, or

F= -VP= -Vpotp, .... (201)

where P is the potential of /o, as denned by (195), and F is as in (198). The slope of P, if by this we understand slope down- wards, or vector rate of fastest decrease, is therefore the same vector as was constructed to solve (197) subject to (199). Taking the divergence of (201), we have, by (197) and (194),

/o = divF= -V2P= -V2pot/x . . . (202) A solution of the characteristic equation of P, or

V2P=-/o, (203)

is therefore (195), and it is the solution vanishing at an infinite distance from the sources.

If we start from (203), we should first use (194), and make it

div(-VP) = /> (204)

Then, by (190), we see that - VP has no curl, so that we have again the two equations (197), (199) to consider, as above.

The consideration of F rather than of P has many advan- tages for purposes of reasoning, as distinguished from calcula- tion. This is true even in statical problems; for instance, when F is electrostatic force, and P the corresponding potential. When we proceed further, to kinetic problems, when F can no longer be wholly expressed as the slope of a potential, the utility of considering P at all, even for calculating purposes, becomes sometimes very questionable, and the consideration is sometimes certainly useless and misleading.

From (202) we see that - v2 and pot are reciprocal. In another form, -V~2 and pot are equivalent; or (pot)"1 and

• v2 are equivalent. The property has only been proved for a scalar function, having a scalar potential. But since any vector 0 may be written iC1 +jC2 + kC3, and the property is true for the three scalars Clf &c., it is also true for the vector

0. Thus, explicitly,

• V2 pot 0 = - V2 pot (iCi + jC2 + kC3) = i ( - V2pot CJ + j ( - V2pot CJ + k ( - V2 pot C3),

206 ELECTROMAGNETIC THEORY. OH. III.

because the reference vectors i, <fec., are constant vectors. So, if Aj is the potential of C15 A2 of C2, and A3 of C3, which makes A be the potential of C, according to (196), we shall have

A = potC, (205)

-V2A = C, (206)

-V2potC = C (207)

In short, the characteristic equation of the vector A merely unites, from the above point of view, the characteristics of the components, so that pot and - V2 are reciprocal when the operand is a vector, as well as when it is a scalar.

Connections of Potential, Curl, Divergence, and Slope. Separa- tion of a Vector into Circuital and Divergent Parts. A Series of Circuital Vectors.

§ 134. But the above gives a very partial and imperfect view of the general theory of potentials. There are numerous other relations between a vector and its associated functions. For instance, if in (206), A be circuital, then the Laplacean y2 may, by (193), be replaced by -(curl)2. That is, if Ax be cir- cuital, and be the potential of C15 then,

curl2 A! = 0!, . . . . (208) or curl2 pot C^Cj (209)

Here, then, we have replaced the scalar operation - V2 by the vector operation curl done twice. Of course, Cx is also cir- cuital, as is proved by (189).

Again, let the A of (206) be polar, or wholly divergent, and be now called A^ the potential of C2; then, by (193), we shall have

-VdivA2 = C2, . . . . (210)

or -VdivpotC2 = C2 (211)

Here again we have replaced V2 by a double operation, first div and then V. This is similar to the passage from (203) to (204), only done in the reverse manner. In (210), by (190), C2 is polar, because A2 is.

Conversely, we see that the potential of a circuital vector is also circuital, and that the potential of a polar vector is also polar.

ELEMENTS OP VECTORIAL ALGEBRA AND ANALYSIS. 207

Now Aj has no divergence, so it may be added on to the divergent A2 in (210) without affecting its truth. Thus, if A = Aj + A2, we have

-VdivA = C2 ..... (212)

Similarly, A2 has no curl, so may be added to the Aj in (208), giving

cur!2A = C1 ...... (213)

Here remember that A is the potential of 0 or Oj + 02.

These equations supply one way of effecting the division of a vector A of general type (having both curl and divergence) into two vectors, one of which, AI} is circuital, whilst the other A2 is polar. For A2 is the potential of 0.2, so, by (212),

-potVdivA = A2 . . . . (214)

separates A, from A. Similarly, Al is the potential of C1? so by (213),

pot curl2 A = Aj .... (215)

separates Ax from A, and therefore A2 from A by a different method. There are many other ways of splitting A into cir- cuital and divergent parts. The one most easily understood, apart from the mathematics, is the following. Go over the whole field of A and measure its divergence. If we find that there is no divergence, then we do not need to go further, for we know that A is circuital already; that is, A = A1} and A2 = 0. But should there be divergence, say B2, so that

divA = divA2 = B2, .... (216)

then construct the flux A2 corresponding to the divergence B2 according to the method already explained with respect to (197); thus,

= -VpotB2, ..... (217)

by (198) and (200). Knowing A2, we know A1? or A - A2.

Or we might vary the process thus. First measure the curl of A. This is the same as the curl of Al because the curl of A2 is zero. Let, then,

B1 ..... (218)

208 ELECTROMAGNETIC THEORY. CH. III.

and construct the circuital solution of this equation ; that is to say, regarding Bx as given, find Ar It is given by

A! = curl pot B! (219)

For Aj as thus defined is evidently circuital, in the first place ; and next, by taking the curl, we produce

curl AJ = curl2 pot B^Bj, . , . (220)

which is the given datum. Here we use curl2 pot = l, because the operand is circuital, as in (209). But instead of (219) we may write

A1=potcurlB1, . . . . (221)

showing an entirely different way of going from Bx to Ar For A! as thus constructed is circuital ; and, since curl Bj = Gv (221) is the same as

A1=pot Op

which was our definition of Al in terms of Cr

Thus pot curl and curl pot are equivalent when the operand is circuital, as above. They are, however, also equivalent when the operand is general, or both circuital and divergent, because if any divergent vector be added to the Bx in the right members of (219) or (221), the operation of curl to which it is subjected renders its introduction inoperative. We therefore have

pot curl C = curl pot 0, .... (222)

where 0 is any vector. We have also the similar exchange- ability,

pot curl2 C = curl2 pot 0, .... (223)

where C is any vector. For, either way, the result is the circuital part of C, or Cr

These results, though puzzling at first from their variety, are yet capable of being brought under rapid- mental control by bringing them together in a compact form. Thus, start with any circuital vector Ar Let Bx = curl Ap Ox = curl Bp Dj =cur!01, &c. We have a series of vectors

Aj, BI} Op Dp Ep . . .

which are all circuital, and any one of which is the eurl of the

ELEMENTS OF VECTORIAL ALGEBRA AND ANALYSIS. 209

preceding. We thus pass down the series one step at a time by means of the operation of curling ; for example,

D^curlOi • (224)

If, however, we wish to go down two steps, we do not need to go first one step, as above, and then another, also as above ; but can make a double step in one operation by means of the Laplacean v2. Thus,

-VSB^Dj (225)

Now go the other way. If we wish to rise up two steps, we can do it in one operation by potting ; thus,

B^potDi (226)

If we wish to go up only one step, we may do it by (224), (225), (226) combined ; that is, either go down one step first, and then up two, as in

B1 = pot Dj = pot curl Gl-) ... (227) or else, first go up two steps and down one j thus,

Bx = curl A! = curl pot Or . . . (228)

There are other less important combinations. But if we wish to make one step up directly, without making use of the double step, we must do it by the Amperean formula, already used, whereby we pass direct from electric current to its magnetic force, which, in rational units, is (when applied to any pair of neighbours Gl and Dx in the above series),

C1 = 2(VD1r1)/47rr2, .... (229)

where i^ is a unit vector from the element D1 to the point at distance r therefrom, where G1 is reckoned. We have now a complete scheme, so far as the circuital vectors are concerned.

A Series of Divergent Vectors.

§ 135. Deferring temporarily a vectorial proof of the last formula (229), which is the only unproved formula in the con- nections of the series of circuital vectors, it will now be convenient to bring together the connections of the divergent vectors and associated quantities. We saw the advantage of the systematic arrangement of the connected circuital vectors

210 ELECTROMAGNETIC THEORY. CH. III.

to be like producing a harmonious chord out of apparently disconnected tones. The advantage is much greater in the divergent series, on account of the less uniform relations in- volved and the greater need of a system to bring them under rapid mental control. In the circuital series, four kinds of operation were involved ; but in the divergent series there are six. The chord will be found to be perfect, though of greater complexity. Thus, let

A2, B2, C2, D2, E,, . . . .

be a series of vectors and scalars connected as follows : — Start with A2, which is to be any divergent vector ; that is, havinr no curl. Let B2 be its divergence ; 02 the slope of B2 ; D2 the divergence of 02 ; E2 the slope of D2, &c. Then A2, C2, E2, . . . are all divergent vectors. But they are separated from one another by two steps instead of one, as was the case in the circuital series last treated. The intermediate quantities are scalars. Instead, also, of the single operation of curl which suffices, in the circuital series, to carry us from any vector to the following one, we now have two distinct opera- tions ; viz., that of slope, when we pass from a scalar to the next vector, as in

C2=-VB2; (230)

and that of divergence, when we pass from a vector to the next scalar, as in

D2 = div02 (231)

But if we wish to go down two steps at once, we can do so by means of the Laplacean operator, whether the operand be a scalar, as in

-V2B2 = D2, (232)

or else a vector, as in

-V2C2 = E2. ...... (233)

In this respect, then, we have the same property as in the circuital series.

We have also identity of operation in going up two steps at once, whether from a scalar to the next higher scalar, as in

B2 = potD2, (234)

ELEMENTS OF VECTORIAL ALGEBRA AND ANALYSIS. 211

or from a vector to the next higher vector, as in

C2 = potE2; (235)

any member of the series being the potential of the second after, as in the circuital series.

Next, to go up one step only, we may utilise the preceding in two ways. First go up two steps and then down one, as in

B2 = divA2 = divpotC2, . . . (236)

where we pass from the vector C2 to the scalar B2 by pot first (up two steps), and then by div (down one step) ; and also as in

02= -VB2= -VpotD2, . . . (237)

where we pass from the scalar D2 to the vector 02 by pot first (up two steps), and then by - y (down one).

Or, secondly, we may first go down one step and then up two, as in

B2 = potD2 = potdiv02, . . . (238)

when rising from the vector C2 to the scalar B2 ; or as in

02 = potE2= -potVD2, . . . (239)

when rising from the scalar D2 to the vector C2.

Finally, if we wish to rise up one step at once, without using the double step either up or down, we can do it by means of

C2 = (div)-1D2 = 2r1D2/47rr2, . . (240)

when we rise from a scalar D2 to a vector 02, which is, in fact, the fundamental formula of the inverse- square law upon which our potential investigations are based. But in rising from a vector C2 to the scalar B2 just above it in the series, we require to use a different process, namely,

B2 = (-V)-1C2=-2r102/47rr2. . . (241)

In these formulas (240), (241), rx is a unit vector from the element in the summation towards the place of the resultant ; that is, from D2 to C2 in (240), and from C2 to B2 in (241).

We now have a complete scheme for the divergent vectors as we had before for the circuital series. On comparing them we see that they are alike in the double steps, either up or down, but differ in the single steps. There is but one kind of

p2

212 ELECTROMAGNETIC THEORY. CH. III.

step up and but one kind down in the circuital series, whereas there are two kinds up and two kinds down in the divergent series. The down step in the circuital series is always done by curl; the up step, shown in (229), may be denoted by (curl)"1. In the divergent series the down steps are done by - v and by div; their inverses may be denoted by (-v)-1 and (div)-1. It is now the nature of these inverse operations (229), (240), and (241) that remains to be elucidated vectorially. The first is the Amperean formula rationalized, whereby we rise from electric current to its magnetic force ; by the second we rise from (for instance) electrostatic force to the electrostatic potential ; or, with a slight change (of sign), from intensity of magnetisation to magnetic potential ; in the third we rise from (for instance), electrification to electrostatic displacement.

The Operation inverse to Divergence.

§ 136. Let p and q be scalar functions, and consider the space-variation of their product. We have

.... (242)

a formula not previously used, but which is seen to be true by observing that it is true for each of the three components of V. Now integrate through any region. We know that

2Np2 = SV(p2), ..... (243)

if N is the unit normal outwards from the boundary of the region, so that the left member is a surface-summation, whilst the right member is a volume-summation throughout the region bounded by the surface. Equation (243) is, in fact, a case of (157), with pq substituted for p. If, then, the surface- summation vanishes, we shall have a simultaneous evanescence of the right member of (243), and therefore, by (242),

2pVq=-2qVp. ...'.. (244)

All the work done by a vector-analyst is exhibited in (244) itself, viz., the transfer of the symbol v from one operand to the other with change of sign, converts the integral of pvq into that of -gvp. The previous remarks contain the justification of the process,

ELEMENTS OP VECTORIAL ALGEBRA AND ANALYSIS. 213

Now - vq is a polar or divergent vector, so may be any one of our divergent series, say C2, when q itself becomes B2. Then

2^C2 = 2B2V^. ..... (245)

Lastly, let p have the special value l/4;rr ; then (245) is the same as

potC2 = A2 = 2B2V^, .... (246)

which exhibits the divergent vector A2 in terms of its diver- gence B2. It is the same as (240), since v# — ij/lnr2, if rx is the nnit vector from B2 to A2.

The Operation inverse to Slope.

§ 137. Next, substitute for q in pq a vector, say g. The new quantity pg has, being a vector, both curl and divergence^ in general. Considering the latter first, we have

div^?g =p div g + gVp, . . . (247)

which is an example of (160). Integrating throughout any volume, we have

SNpg = 2divjpg, .... (248)

as in (159), where N is as before. So, if the surface-integral vanishes, we obtain, by (247),

2^divg= -SgV^, .... (249)

and, in this transformation, all the vector-analyst has to do is to shift the operator v from one operand to the other, and change the sign.

The vector g here has no restriction imposed upon it. It may therefore be of the general type 0 = 0! + C2, giving

2^D2 = - 2 GVp = - 2 G2Vp . . . (250)

Here the portion 2 Gl Vp vanishes because Cj is circuital and yp is polar, which is one of the important theorems in analysis that become visibly true by following the tubes of the circuital flux in performing the summation, when the summation is seen to vanish separately for every tube. (See § 87.)

If in (250) we give^? the special value l/4?rr, viz., the poten- tial due to a unit source at distance r, we obtain

pot D2 = B2 = - 2 CW = - 2 G2Vp, . . (251)

214: ELECTROMAGNETIC THEORY. CH. III.

showing how to pass up one step in the divergent series from a vector to the preceding scalar. It is the same as (241), remembering the value of Vp.

The Operation inverse to Curl.

§ 138. Thirdly, we have the curl of pg to consider. Here, by (188),

curing =#> curl g- VgVp, . . . (252)

Integrating throughout any region, we obtain

2VN^g = 2curl^g, .... (253)

which is a case of (162), with pgput for H. So, if the surface- summation vanishes, we obtain, by (252),

2^curlg = 2VgVp, .... (254)

where the symbol VV is moved from g to p, with a change of sign, as before. In this, take £> = l/47rr; then, since there is no restriction upon g, we get, taking g = C,

. . (255)

which is the companion to (251), showing how to pass up one step in the circuital series, from Ox to Br This is equivalent to (229). The divergent part of 0 contributes nothing. That is to say, for example, the magnetic force due to a completely divergent distribution of electric current, according to Ampere's formula for the magnetic force of a current element, is zero. We might, indeed, argue from this, that there could not be such a kind of electric current ; that is to say, that the current must be circuital, since the mathematical machinery itself* constructed on old ideas, automatically rejects the want of circuitality, and refuses to admit the purely divergent part as contributory to magnetic force. This is a perfectly valid argument, provided the test of the existence of electric current be the existence of magnetic force, which is tantamount to what Maxwell insisted upon, in another form.

For instance, if we calculate by (255) the magnetic force due to a supposititious current element at a point, simply by re- moving the sign of summation, we obtain the magnetic force of a rational current element, a system of circuital current resem- bling the induction due to a magnetised particle. (See § 62.)

ELEMENTS OP VECTOIUAL ALGEBRA AND ANALYSIS. 215

Remarks on the inverse Operations.