298 



NATURE 



[July 30, 1891 



is increased to F + dY , e-^. 



become e^ + de-^. 



e^ + </i?2, ^3 + de^y the work done by the stresses will, 

 neglecting small quantities of the second order, be 



KF 



Stt 



dv ( dfe] — de^ — de^ ; 



and if the increase of potential energy in the element 

 take place in consequence of the work done against the 

 stresses we get the equation 



\-dv {de^ - de^ - de^ = "^^dv, 



2dY 

 F 



de-^—de^-de^ 

 which gives by integration 



^1—^2-^3=2 log F + const. 



This result is inadmissible, since when F is zero, we 

 must have e^ = e.^ = ^3=0, while if this equation holds 

 either e,^ or e^ is infinite. 



A solution of the difficulty is simply that the energy is 

 not really potential but kinetic. It is certainly not easy 

 to see why the electro-magnetic energy should be regarded 

 as kinetic and the electro-static as potential, and it seems 

 more natural to conclude, as all progress in knowledge 

 of matter seems to indicate, that the properties of the 

 medium are wholly due to motion. 



After a short sketch of purely magnetic theory, Poin- 

 card proceeds to what must be regarded as the most 

 important part of his account of Maxwell's work — the 

 theory of electro-magnetism. His investigation of the 

 magnetic potentials of circuits is somewhat different 

 from that usually given. Maxwell takes as his starting 

 point here the equivalence of a current-carrying circuit 

 of small dimensions and a magnet. Poincar^ bases his 

 method directly on the following three results of experi- 

 ment : (i) that two parallel currents of equal intensity 

 and of opposite directions in two close conductors exert 

 no action on a magnetic pole at some distance ; (2) if one 

 of these currents have small sinuosities, its action on the 

 magnetic pole is still equal and opposite to that of the 

 straight current ; and (3) that the magnetic action is pro- 

 portional to the quantity of electricity which traverses a 

 cross-section of the conductor in the unit of time. 



With the assumption that the components of the force 

 acting on a magnetic pole are obtained by partial differen- 

 tiation of a function which depends only on the relative 

 positions of the pole and the circuit, the usual theorems 

 are obtained in the following elegant manner. First of 

 all it is shown that the potential of a closed plane circuit 

 at any point in its plane is zero. This is first proved for 

 a circuit symmetrical about a line on its own plane and a 

 point on the axis of symmetry. Then by using the first 

 fundamental proposition to introduce across the circuit 

 straight conductors each carrying two equal and opposite 

 currents equal to the current in the circuit, a circuit of any 

 form is divided into narrow portions each bounded at the 

 ends by elements of the circuit, and at its sides by radial 

 lines passing through the point in question. By using 

 then the second proposition to replace each end-element 

 of the circuit by a circular arc passing through the centre 

 of the element and described from the given point as 

 centre, each strip is turned into a complete circuit, sym- 

 metrical about a line through the given point. Since, 

 then, the theorem is true for every such circuit, it is true 

 for the whole given circuit which they build up. Next it 

 is easily shown that when a circuit is situated on the 

 surface of a cone but does not surround the axis— that is, 

 is such that a generating line meets the circuit in an even 

 number of points— the potential of the circuit at the vertex 

 of the cone is zero. For, by means of conductors intro- 

 duced along generating lines, and carrying equal and 



NO. I 135, VOL. 44] 



opposite currents as before, it is possible with the aid of 

 the second result stated above to replace the circuit by a 

 number of narrow plane circuits each carrying the given 

 current, and symmetrical about a generating line of the 

 cone. Hence each element produces zero potential at the 

 vertex, and therefore so also does the given circuit. 



Then it is proved that two circuits on the surface of a 

 cone, each passing round the axis, produce equal and 

 opposite potentials at the vertex, if the currents are 

 equal and flow in opposite directions round the cone. 

 For by mean^ of hypothetical conductors introduced as 

 before along the generating lines, and the second funda- 

 mental result, these circuits can be converted into narrow 

 plane circuits, each carrying a current and symmetrical 

 about a generating line. Thus the arrangement of two 

 circuits produces no potential at the vertex. It is to be 

 observed that the two circuits subtend equal solid angles 

 at the vertex of the cone, and that the potentials must 

 still be equal and opposite if the circuits surround 

 distinct superposable cones. 



Considering now any closed circuit, we can draw a 

 cone from any chosen point as vertex, so that the genera- 

 tors pass through the circuit. Then this cone can be 

 divided into an infinite number of infinitely small super- 

 posable cones of equal solid angle, each having a 

 current flowing round it in the same direction as that 

 round the given circuit, and the total potential at the 

 common vertex is the sum of the equal potentials pro- 

 duced by three small circuits — that is, the potential is 

 proportional to the solid angle subtended at the point by 

 the circuit. 



The equations connecting the components u, v, w, 

 of currents with the components of magnetic force and 

 magnetic induction, the relations connecting the mag- 

 netic force and magnetic induction, those connecting the 

 magnetic force with the vector potential (which Poincare 

 calls the moment ^ledromagnHique), and the value of the 

 components of the latter quantity for a linear circuit with 

 their application to the proof of Neumann's expression 

 for the " electrodynamic potential " (the mutual intrinsic 

 energy) of two linear current-carrying circuits, and the 

 corresponding expressions for the " electrodynamic poten- 

 tials " (electrokinetic energies) of the circuits themselves, 

 are dealt with in the next two chapters. 



In chapter ix. we come to the most important part of 

 the book, the theory of induction, and the treatment 

 of this part of the subject is instructive. It is a result of 

 experiment that if the currents y^, y^, in two fixed cir- 

 cuits C^, C,, respectively, are varied, electromotive forces 

 Adyxjdt + ']idy2ldt, ^dyjdt -\- Cdy^ldt are produced, where 

 B is a coefficient depending on the relative positions of the 

 circuits, A a coefficient depending on Ci alone, and C a co- 

 efficient depending on Cg alone. Thus if the circuits are 

 deformed or relatively displaced, electromotive forces of 

 amounts y-^dhldt -\- y.^d^ldt. y^dBldi -{- y^dCjdt, are pro- 

 duced in Ci and C2, so that the total electromotive forces are 

 respectively d{k.y^ -\- ^y^ldt, and di^y^ -\- Qy^dt. Now 

 by the circuits, in which are supposed to act impressed 

 electromotive forces Ei, Eg, the energy furnished in time 

 dt is Eiy^dt -\- Y.^y^dt. This must be expended in heating 

 the conductors, and in doing all the work which is done 

 in the displacement or deformation of the conductors. 

 This latter work is of two parts, (i) that which is done 

 in consequence of the geometrical alteration of the 

 circuits, (2) that which is done in virtue of the change of 

 the current strengths. But the " electrodynamic poten- 

 tial " of the system (Maxwell's electrokinetic energy) is 



T = \{l^,y^ + 2M7iy2 + Uy.f), 



so that the former work is 



aT = \{y^'dU + 2yiy,^M -f yi^dl.^. 



Thus the work ^W done in virtue of the changes of the 

 currents is the difference between this and the excess of 



