368 



NA TURE 



[February i8, 1892 





(2) The action of a closed circuit carrying a current 

 upon any current element is normal to the element. 



(3) The action of a closed (non-varying) solenoid upon 

 a current element is zero. 



It is besides assumed that the action of a circuit upon 

 a current element is the sum, in the dynamical sense, of 

 the individual actions of the elements of the circuit ; and 

 that the action between two elements is a force in the 

 straight line joining their centres. 



The process used for the deduction of Ampere's formula 

 from these premisses is very elegant. If ds, ds' be the 

 lengths of the two elements, y, y' the currents in them, 

 e the angle between the elements, B, Q' the angles they 

 make with the line joining their centres, the action of ds 

 on ds' may be represented hy f{r, 6, 6', f)yy'dsds'. But 

 the action of ds may, by the first principle stated above, 

 be replaced by the actions, of its components dx, dy, dz ; 

 so that 



/= A^ + B^-^ + C^^ 

 as ds as 



where A, B, C are coefficients. Now, / depends upon 

 r, 6, 6\ f ; r and ^'do not depend on the direction cosines 

 of ds ; cos 6 and cos e are linear and homogeneous 

 with respect to these direction-cosines. Hence / must 

 be linear and homogeneous with respect to cos 6 and 

 cos e, that is with respect to drjds, and d^rjdsds'. Simi- 

 larly, /is linear and homogeneous with respect to drjds', 

 d'^rjdsds'. Hence we have 



/- , , ^ dr dr , ^, ^ d'-r 

 ^^'^^'-^s ds'-'^^^'^dldP' 

 where -^{r) and (p{r) are functions of r. 



These functions are determined by the second and 

 third fundamental principles. The second gives yj/if^) 

 = (t>'{r), so that the problem is reduced to the determina- 

 tion of <^(r). This value of yfr{r), however, permits / to 

 be written in the form 



2dU/dr . d-^U/dsds', 

 where U is a function of r only, and 

 dU/dr = V^(r}. 

 From this it is then shown that, if T be the so-called 

 electrodynamic potential (electrokinetic energy) of the 

 circuits — that is, the function the space variation of 

 which, for any direction, is the force in that direction 

 between the circuits — 



^ f fdU dU 

 J Jds ds' 



the currents being each unity, and the integrals being 

 taken round the circuits. 



The determination of U is then effected by means of 

 the third principle. It is first shown that T may be 

 written as the integral of Fd.r -^Gdy + Hdz round the 

 circuit to which ds belongs, F denoting the integral 

 round the other circuit of U'-ds'dr/ds' . {x — x')lr, and 

 G, H similar expressions. F, G, H are, in this theory, 

 what Maxwell has called the components of vector 

 potential. These values of F, G, H, it is to be remarked, 

 fulfil the relation 



dFjdx + dGjdy + dYijdz (= J) = o. 



By applying the third principle it is proved that, if 

 V- have its ordinary signification, and /'(r) = U'-/^, 

 (U' = dUjdr), V'f{r) must be a constant, in order that 

 the action of a closed non-varying solenoid on a com- 

 plete circuit may be zero. Since f(r) must be zero at 

 infinity, this gives/(r) = kjr^ and if the ordinary electro- 

 magnetic definition of unit current be taken, /' must 

 be unity, so that U' = ± ij ^r. Hence the attraction 

 between the elements is 



2-yy'dsds' - (cos e - f cos d cos ff), 

 Ampere's well-known expression. 

 NO. I 164, VOL. 45] 



The above expressions for F, G, H reduce easily to 



\/{r)dx', \f{r)dy', \/{r)dz', so that, putting in the 



value of /(r), we get the well-known value of the mutual 

 energy of the two circuits — 



T=ff^-^'dsds'. 



The theory of induction is next taken up. After a 

 short discussion of some objections made by M. Bertrand 

 to the received method of deducing the laws of induction 

 from the observed facts of electromagnetism, M. Poin- 

 card proceeds to show that the electrokinetic energy of 

 two currents is equal to the electrodynamic potential, 

 and recalls Maxwell's application of Lagrange's dyna- 

 mical equations to the theory of inductive action. He 

 then deals at some length with the celebrated theory put 

 forward by Weber for the action between two quantities^ 

 e, e', of electricity, as depending on their distance apart 

 and their motion. 



This we pass over, with the remark that Poincard here 

 discusses certain difficulties to which the theory leads in 

 connection with the value it gives for the action between 

 two current elements, and concludes with a short analysis 

 of Maxwell's examination of the theory of induction as 

 deduced from Weber's law. According to Maxwell 

 (" El. and Mag.," vol. ii. p. 445, second edition), 

 Weber's theory gives, for the inductive electromotive 

 force exerted by the circuit in which the current y flows 

 on the other, the equation — 



^J [yL±±^dsds', 



dtj J r ds ds' 



whichj for a closed circuit, agrees with the well-known 

 equation— 



M. Poincard points out that this apparent agreement 

 of the two theories is due to the fact that Maxwell has 

 overlooked certain terms which contribute to the value 

 of E, and which do not give a zero result when integrated 

 round a closed circuit. 



The expressions given by Weber and Neumann for the 

 mutual potential of two current elements are next con- 

 sidered, and shown to be included in the general expres- 

 sion given for the same potential by Helmholtz. By 

 means of this expression Helmholtz's general electro- 

 dynamic theory is introduced, and then follows an 

 elaborate comparison of the theories of Helmholtz and 

 Maxwell. It is shown that Helmholtz's theory leads to 

 the value of T for conduction in three dimensions given 

 by the equation — 



T = J i{¥ti + Gv + ll7v)d?Si, 



where d^ is an element of volume, ti, v^w the components 

 of currents, and the integral is extended throughout all 

 space. F, G, H, are, of course, the components of vector 

 potential, and in this theory are given by equations — 



where 



F= i y'dx'lr + i(i - k)d^ldi 



'x, &c. 



jy'ds' 



drjds'. 



Ifp be the density of free electricity at any point, 

 dujdx + dvjdy + d-MJdz = — dp/d/, 

 and this, when instead of y'dr/ds' is substituted its value 

 in terms of u', v', w', gives, by an application of Green's 

 theorem, the result — 



,;, = Irdp'/d/ . aiS', 



where da' is an element of tha space in which the current 



