Electromagnetic Theory, Vol. 2 (1899) — part 14 of 31

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

2ourlH-f 2VnH-0. .... (163)

If H be magnetic force, curl H is the electric current-density io the region. Now YnH 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 ; ai^ by taking it to coincide with H, so as to make the circuitation a maximum, we find that YnH represents the surface-density of ouirent. So, if J be current, we have, by (163), 2 J = 0 for any region, by itself. The surfaoe-distribation and the volume-

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XLEMENI8 OF VECTORIAL ALGEBRA AND ANALYSIS. 197

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

iiVVH--VVnH, .... (164)

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

live Examples of the Operation of V in Transforming firom Oircnital to Suxface Summations.

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

2F(T) = 2/, (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 normalB are i, j, k, we readily see that the corresponding /'s are

V^(k)- V.F(J), v,F(l)-v,F(k), v,F(J)-7^(l).. (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 vectorSi that the sum of any vectors forming, when put end to end, a eircuit, is «ero.

(&). Take F(T) » TP, where P is a scalar function of positioiL Then, with normal 1, we have, by the first of (166),

ycVakP-VajP^ViV.P; . . (168) •0^ writing N for i, we obtain

i:TP-:^YNV.P-2VNV.P . . . (169)

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198 ELECTROMAGNETIC THEORY. CH. III.

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

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

/- VJsH - VJH - VjHs - VgHj-iWH,

by (149). Therefore, putting N for i,

2TH-2irVyH»2HoarlH . . (170)

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

STH-2VNVA .... (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 (106) gives

/- VaVkH - VjVjH = V (kV, - j Vg)H. But here we have kVj - j V3 = ViV, 80 that /«V(ViV;H5 and therefore, generally, putting N for i,

2VTH-2V(VNV)H. • . . (172)

(«). 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)-(VTK)P.

We then find, taking N = i, and using the first of (166),

/ = VsVkl . P - V3 Vji . P - (j Va + kV8)P - VP - i (i V) P.

In general, therefore,

2VTN.P-2(VP-H.NVP)-2V.P-SV(VNV)H.P .(173)

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

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ELEMENTS OF VECTOUIAL ALGEBRA AND ANALYSIS. 199

may have out of the surface. The last form of (173) inyolyes the transformation formula (52). Obserre that in all the above examples,

2F(N)-2F(V) (174)

when we pan from a closed surfaoe to the enolosed rsgton; and that

2F(T)-2F(VNV), .... (176)

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 Esnmples of the Differentiating Effects of V*

§ 13Z The following examples relate principally to the modi* fioatione 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 is not open to misconception. But in the second form, expressing the divergence of YEN, since TX ioUows V, we must understand that N is supposed to remain constant. In the third form, again, the operand E precedes the differenUator. We must either, then, assume that y acts back- wards, or else, which is preferable, change the third form to YN V.Bi the scalar product of VNv and E ; or (VKv) E^ if that be plainer.

(6). Suppose, however, that both vectors in the vector pro- dnet axe variable. Thus, requured the divergence of YSH, expanded vectorially. We have

VVEH-EVHV-HVVB. . . . (177)

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■LBOTROMAGNBTIO THIOBT.

OB. m.

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

div VEH = HcurlE-EcurlH, . . . (178)

A very important transformation. It is concerned ia the de- doction 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 iu the other. Taking the former, we have

curl VqB-VVVqB, .... (179)

where B is the induction and q the velocity of the medium supporting it. Apply the elementary trausformatioa (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 penoM 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 V, when q alone, and Vb when B alone suffers differentiation. Then, fully,

WVqB = VV,VQB + VVBVqB, . (182)

VV,VqB = BV.q~Bdivq, .... (183)

Wb VqB = q div B - qV.B (184)

SL£>1J£M^ OF VECTOniAL ALGEBRA AND ANALT8I8. 201

Hm the sum of (183) and (184) gives (181). The mean- ing 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 - Oq) = 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 vrcr. Further, we have B + av.B»SB/5^, by (127), §122, so that (185) becomes

-onrl(S-eo)»K+^/5<+BdiYa-BV.q, . (186)

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

curl (H - ho) = C + 5D/Si + D div QL - Dy.QL. . (187)

The time-Tariations refer to the same (moving) portion of the medium now. But if we wish to indicate the movement of electrification, (&o., through the medium, that is, have relative motion n - q of p (and w - q of cr) with respect to the ether 4)hen to the right side of (187) add the term {n-q^p, and xo the right ride of (186) add (w > q)<r.

It is desirable to preserve the velocities n 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 -scsdar and vector under consideration. Now examine its curl. Thus,

curlDP = VV(DP)

-iVVD.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, 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 VVVH-0. . . (189)

BLlOTBOlf AONKIG THIOBT.

OH. IIL.

If V * Teetor (189) woald mean tbat tii»

volume of a parallelepiped yanishes when two edges coincide.

(ff), A somewhat limilar oaae it presented by the Taniahmg' of the curl of a polar force. Thns,

curlVP«=0, or VV VP-0. . . . (190)

Of oourse Yvy is zero. But the soalar product W, or v*, ia the Laplaoean operator,

Va-Vi2 + V,2 + Ve«, . • . . (191) which occurs frequently.

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

(curl)«A-WWA

= V VA-V2A, . . . (192)

by the transforming formula (52). Or

V-A = V div A - curl^A. . . . (193)

Thus there are two principal forms. If the vector A has no- curl, then v^A is the slope of its divergence. If, on the other hand, it has no divergence, then - has the same effect ezaotly as taking the curi t^ice.

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

V«P-divVP, (194)

the divergence of the slope of the scalar.

The Potential of a Scalar or Vector. The Ohanusteristie Eguation 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 defining the potential at A of a quantity p at B to be the quantity p/47rr, 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*

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BIJEMENT8 OF YSCTORIAL ALGBBRA AND ASALYSO, 203

point is the sum of the potentials of all the elements of /»• That is,

P-pot/>-2p/4«r, .... (195)

if P is the potential of p.

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

A-pot0-20/4irr. .... (196)

The sumouktion is now a veotor 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 unnecessazy to say so, but one oannot be too partionlar.

We may conneot these potentials with v as follows : — Gi^en that

divP-ft (197)

that is to say, that the divergence of a vector F in 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, nnce it is derived from F by differentiation, which should, perhaps, be regarded as a direct process, rather than inyeise. But if it be p that is given, F is not immediately determinable, nnlees 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/Ain4 is

the intensity of the " force " at distance r from the point-source p, and T^pj^irr^ the vector force to correspond, where Tj is a unit vector drawn from p towards the point under considera- tion, making the resultant F be

P-20>/4»f*)ri,

(198)

■LKCTROlfAONKTIO THEORY.

CH. m.

we obtdn a special solution of (197) which has a remarkabls

property, viz.,

so that if F be electric force, the Toltage in any oironit is wm. The meaning of ciirl, I may obeenre| has been explained more than once ; both its intrinsic and its Tectorial 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 th^ 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 (lOS) 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, P 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 oirouitaL So f is non- existent.

Now observe that

or the slope of the scalar l/i-rrr is the vector with tensor l/4aT* and direction r,. It follows from this that

curl F i» 0,

(199;

-V(l/47rr) = ri/4irr«,

• V pot /i = F,

(200)

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ELEMENTS OF VEOrORIAL ALG^RA AND ANALYSIS. 20d

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

P- -VP- -Vpot/3, .... (201)

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

p-divP- -V«P- -V2 pot/a. . . . (202) A solution of the characteristic equation of P, or

V«P--/» (203)

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

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

div(-rP)=/) (204)

Then, by (190), we see that - VP has no curl, so that we have •gain 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 tme 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 -v* and pot m reciprocal. In another form, -V* and pot are equivalent; or (pot)~i and

• V* are equivalent. The property has only been proved for a scalar function, having a scalar potentii^. But since any vector C may be written iCj+jCg + kCg, and the property is true for the three scalars C^, <&c., it is also true for the vector

0. Thus, explicitly,

• V*potO« -V«pot(iCi+JC,+kCR) -1 ( - V^pot Ci) + j ( - ?2pot + k ( _ y2 pot CX

206

SLBQVfiOllAOKBnO THlOEr.

OH. m.

beoMue the vafereno^ Yeoton U ^ ooofbuit Teoton. So^ if A| is the potential of C^, A, of C,, and A,of Cg, which makes A be the potential of according to (196), we shall have

A = potO, (205)

-V«A-0| (206)

-^potO-0. (207)

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

Connections of Potential, Onrl, Divergence, and Slope. Separa- tion of a Vector into Oircnltal and BiTergent Ftota 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 Laphicean may, by (193), be replaced by - (curl)^. That is, if A^ be cir- •cuital, and be the potential of 0^ then,

curl<A|-iOp .... (208)

or curP pot Oi = 0^ (209)

Here, then, we have replaced the scalar operation - by the vector operation curl done twice. Of course, 0| is also dr- ouital, as is proved by (189).

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

-VdivA,-0^ .... (210)

or - VdivpotO, = 02 (211)

Here again we have replaced by a double operation, first ' div and then 7. This is similar to the passage from (203) to (204), only done in the reverse manner. In (210), by (190^ €2 is polar, because A, is.

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

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KLEMBNTB OF VECTORIAL ALGEBRA AND AKALT8I3. 207

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

-VdivA = 0, (212)

SimiUriy, A, lias no oarl, bo may be added to the A^ in (208), giving

curlSA^O, (213)

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

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, Aj, is circuital, whilst the other A^ is polar. For A^ is the potential of 0,, so, by (212),

-potVdivA-Ag .... (214)

separates A^ from A Similarly, A| is the potential of 0^, so by (213),

potcurl^A^Aj .... (215)

separates Aj from A, and therefore A.2 from A by a diflerent 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 = Aj, and A^aiO. But should there be divergence, say B^, so that

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

then oonstruct the flux oorresponding to the divergence B^ aooording to the method already explained with respect to <197); thus,

A,«2(Bj^4irr2)ri

--VpotBj (217)

by (198) and (200). Knowing A^ we know A^ or A - A,.

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

curl AbcutI Ai = B| (2 IS)

208

■LBOTBOXAGKRIO TH>OBT.

GH. in.

and ooDStraot the circuital lolution of this eqnatioa ; that Ig to

bay, regarding as given, find A^. It is given bj

A^- curl pot (219)

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

carl Aj»carl< pot B|»B„ . • • (220)

which is the given datum. Here we use curl^potal» because the operand is circuital, as in (209). But instead of (219) we may write

A, = potcurlBi, .... (221)

■howing an entirely different way of going from B^ to A^. For A| as thus constructed is circuital; and, since ourlB|BO|» (221) is the same as

Ai—pot 0|,

which was our definition of A| in terms of 0|.

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 Bj in the right members of (-19) or (221), the operation of curl to which it is subjected renders its introduction inoperative. We therefore have

pot curl 0" curl pot 0, .... (222)

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

potcurl2 0 = curl2potO, . . • . (223)

where 0 is any Tcctor. For, either way, the result is the circuital part of 0, or 0|.

These results, though pnnling at first from their Tariety, are yet capable of being bronght under rapid mental control by bringing them together in a compact form. Thus, start with any circuital yeotor A^. Let curlAj, 0^»ouzl Bi, D, " curlOi, &0, We have a Beriea of vectors

^1 Dj, S|, • • •

which are all cinraital, and any one of which is the tfoil of the

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SLIICBKTB OF TSOTORIAL ALOBBRA AND ANALYSIS. 209

preceding. We thus pass down the series one step at a time

by means of the operation of curling ; for example,

Di«ottrlOj • (224)

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 ;

bat can make a double step in one operation by means of the Laplacean Thus,

• V^Bi = Di (225)

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

Bi = pot Di (226)

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

B| -= pot pot curl Oi; . • • (227)

or else, first go up two steps and down one ; thus,

Bi — ourl A| — ourl pot 0|. . . • (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 Amp^reau 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 0^ and in the above series),

Oi-2(VDiri)/4»r«, .... (229)

where Tj is a unit vector from the element Dj to the point at distance r therefrom, where is reckoned. We have now a oomplete 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 oonvenient to bring together the connections of the divergent vectors and associated quantities. We saw the advantage of the systematic amngement of the connected circuital vectors

210

aLBCTBOHAONETIO THBORT.

CH. in.

to be like producing a harmonious chord out of apparently disconnected tones. The advantage is much greater in the diveigent series, on account of the less uniform relations in- ToWed 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

B], C2, T)p £-2* • • • •

be a series of vectors and scalars connected as follows : — Start with Ao, which is to be any divergent vector ; that is, havinr no curL Let B., be its divergence ; Co the slope of B., ; the diveigence of ; the slope of &c Then A,, O^t E2, ... are all divergent vectors. But they are separated from one another by two steps instead of one, as was tiie 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

0,--VBj; (-230)

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

Dj-divOj. (231)

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

-V2B2 = D, (232)

or else a vector, as in

-V^Oj-Bs. (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

Bs-^potD,, (234)

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ELBMSirra OF VmOBIAL ALGEBRA AHD ANALT8IS. 211

<Nr from a vector to the next higher vector, ae in

0,- pot Eg; (235)

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

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

B^^divAs^divpotOy • • . (236)

where we pass from the vector 0^ to the scalar B., by pot first (up two stops), and then by div (down one step) ; and also as in

-VBo= -VpotDg, . . . (237)

where we pass from the scalar D2 to the vector O2 by pot first ^up two stops), and then by - y (down one).

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

Bj«potDj«potdivO,. . . . (238)

when rising from the vector Oj to the scalar B, ; or as in

Oj = potE,= -potVDjp . . . {'2o[})

when rising from the scalar to the vector O,.

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

0,-(div)-iDj = 2riDj/43rr«, . . (240)

when we rise from a scalar to a vector 0,, 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 O. to the scalar B, just above it in the series, we require -to use a different process, namely,

Bj = (-.V)-^Oj,= -2ri0j/4jrr2. . . (24l)

In these formulie (240), (241), is a unit vector from the ^element in the summation towards the place of the resultant ; that is, from to 0, in (240), and from 0, to B, in (241).

We now have a completo 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 di£fer in the single steps. There is but one kind of

p2

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SUOIBOKAGMBTIG THBOBT.

step up and but one kind down in the oixonitel seriea^ wheraw there are two kinds up and two kinds down in the divergent series. The down step in the oirouital series is always done by enrl ; the up step^ shown in (229), may be denoted by (ouri)~^« In the divergent series the down steps are done by - v and by div; their inverses may be denoted by (-v)-^ and (div)-^ It is now the nature of these inveme operations (229 (240), and (241) that remains to be eluddated veotorially. The first is the Amp^rean formula rationaliaed, whereby we rise from electric current to its magnetic force ; by the second we rise from (for instance) electrostatic force to the electroetatio potential ; or, with a slight change (of sign), from intensity of maguetisation to magnetic potential ; in the third we rise from (for instance), eleotrification 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

V(p2)=pV2 + gVi?, .... (242)

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

2N|^-2V(f>9), (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),

7.pVq^-^qSp (244)

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

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ELEMENTS OF VECTORIAL ALQBBBA AND ANALYSIS. 213

Now - v^r is a polar or divergent vector, so may be any one of our divergent series, say 0^ when q itself becomes B,. Then

2|KI,»2B2V/>. (245)

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

potOs^A^^SB^Vi?, • • • . (216)

which exhibits the diveigent vector in terms of its diver- gcDoe B|. It is the same as (240), since vp'=Tj^irr^, if is the unit vector from to

The Operation InTene to Slope.

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

diypg^pdiYg-hgVp, . . . (247)

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

2N;?g = 2divj9g, .... (248)

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

22?divg= -SgVp, .... (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^ + 0,, giving

2pD,- -20yii--20,Vp. . . (250)

Here the portion 2 Oi'^p vanishes because 0^ is circuital and is polar, which is one of the important theorems in analysis £hat become visibly true by following the tubes of the circuital flux in performing the summation, when the summation is seen to vanish separately for eveiy tube. (3ee § 87.)

If in (250) we give^ the special value l/47rr, viz., the poten- tial due to a unit source at distance r, we obtain

potD,-B,- -20Vi'«-20,V#>, . . (251)

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ELECTROMAGNETIC THEORY.

CH. Ill,

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 V^.

The Operation inverse to CurL

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

•urljpg^p curl g-YgV/?, • • • (252)

Integrating throughout any region, we obtain

^Y^pg^'Zcuiljpg, .... (253)

which is a case of (162), with pg put for H. So, if the eurfiboe- summation Tanishefl, we obtain, by (252),

2j>curlg-2VgVp, • . . . (254)

where the symbol W is moved from g to p, with a change of sign, as before. In this, take p^l/^vr; then, since there is no restriction upon g, we get, taking g = 0,

potDi-Bi-2V0yp-2V0iVi>, . . (255)

which is the companion to (251), showing how to pass up one step in the cireuital series, from 0^ to B,. 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 itselft 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 fovoe 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.)

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ELBICEKTS OF VECTORIAl. ALGBBBA AND ANALYSIS. 215

fiemarks on tbe inverse Operationfl.