462 Messrs. Watson and Burbury on the Law 



canon we have been calling in question. Moreover this law 

 (if not disproved by some decisive experiment) would greatly 

 simplify the mathematical treatment of electrodynamics. 

 Since for two elements 



we have 



V = uii / dsds f , 



r 



dV . ., cos e , , , 



—r- = — im — — as as . 

 dr r" 



For parallel currents in the same direction cose = l; in op- 

 posite directions, cos 6= —1. Therefore, " Parallel elemen- 

 tary currents, if in the same direction, attract each other loith 

 a force varying inversely as the square of the distance ; if in 

 opposite directions, repel each other ivith the same force." 



For perpendicular currents, cose=0; therefore — =0, or 



"Elementary currents at right angles to each other exert on each 



other no attractive or repulsive force ." 



Again, 



dY . . sin e 7 ' -, 



—r = — an as els . 



cle r r 



For parallel currents, sin 6 = 0; for perpendicular currents, 

 sin e=l. Therefore "Parallel elementary currents exert on 

 each other no couple-action!^ 



"Elementary currents at rigid angles to each other tend, to turn 

 each other into a position of parallelism with a force varying in- 

 versely as the distance." : 



The plane in which either element tends to turn is that in 

 which the variation of cos e for a given small angle turned 

 through is a maximum. Resolving the elements into their 

 components, we find that the component idx tends to turn 

 i! dy' round the axis of z, and i' dz 1 round the axis of y ; hence 

 the turning- couple exerted by ids on i r ds' is 



. . C civ dy' __ dy da/ \ 

 ^ % \Ts d7 ds~W'S 



ds ds' 



round the axis of z; and similarly for the other axes. 



18. Every elementary current in a given direction, as idx, 



% dx 

 has at any point in space distant r from it the potential — i 



which being a vector or directed quantity, namely having the 

 direction x, is with propriety termed the vector potential of 

 the element, Maxwell indeed states distinctly (vol. ii. p. 267) 



