by a Magnet in a Rotating Conductor, 517 



from which equations, in combination with (14), the functions 

 L, M, N, V are to be so determined that the surface- condition 



/BM BN\ . , ZdN BL\ , /&L BM\ " , ln . 



shall be simultaneously satisfied. It should here be remarked 

 that, since the equations (13) result from the differentiation of 

 (11), the former are necessary consequences of the latter; the 

 converse of this, however, is not true. The equations (18) 

 therefore express the possibility of a more general motion of 

 electricity than do the original equations (15), so that the former 

 must always be satisfied by integrals of the latter, but not con- 

 versely. Consequently the arbitrary functions involved in any 

 general integrals of the equations (18) will have to be so spe- 

 cialized that the equations (11) may be simultaneously fulfilled. 

 For instance, the integral equations L = 0, M = 0, N=0, which 

 imply also u = 0, v = 0, w = 0, constitute a particular solution of 

 the equations (18) ; not being consistent with the equations (11), 

 however, this solution cannot be regarded as a solution of the 

 equations (15) unless P= const. A system of equations for 

 u } v, w, exactly similar in form to the system (18) for L, M, N, 

 is obtained when, conversely, L, M, N are eliminated from (15) 

 by means of the equations (13). In fact, if by differentiation 



of (15) we form ^ ^ — and the other analogous expressions, 



and put 



¥=--^(uL + bM + foN), 



we shall find for the determination of the functions u f v, w, ^P 

 the equations 



oz oy «■ o% J 



ow Bw . 1V J B^ ; - 1 



s ^- =47r£K 1 — ^ h fow— uw > , 



ox oz ^ oy J 



d M ov a 7xr J B^ , . \ 



^ ~ =47T^KS — ^- +uv— bu > . 



By q-z I B* J ! 



(20) 



QU QV QW 



ox oy oz ' 



to which the same remark applies as was previously made with 

 reference to the equations (18). A knowledge of w, v, w is not 

 necessary when the object is merely to determine the action 

 exerted upon an external magnetic pole by the system of cur- 



