Field of an Electrical Current. 211 



are X, Z parallel to OD, OV, we have 



dX dZ dY = 

 dx dz dy ? 



d\ 

 where -r- is the line rate of increase of force at P perpen- 



dy 

 dicular to the plane, POD, of xz. Y is, of course, zero, but 

 we easily see that 



dY = X 



dy a — m cos (/> 



by taking a position of the point, or pole, close to P along 

 the perpendicular at P to the plane xz. Hence we should 



have, up to and including terms in -g, the result 





d® 





dm 



d® _ 



md(p 



2 

 m 



d® d® 



The values of X and Z are best found from -y- and — — , 



am ma<p 



which (neglecting A for the moment) are the components of 



force along and perpendicular to DP. 



Now 



, v d% , , d® . j ..„. 



also X= — j— cos d>-\ r-rsm®, (47) 



am r rnd(p T J 



„ d® . d® ' 



Z= -j-smdH -jjcosd), (48) 



am mdcp T 



, d t d sin <£. d 



and -^- = —003 0-^ 1- -j- ±i 



ax am in a<p 



d , d cos <b d 



-j-= sine/)— - + T -r,' 



dz dm m dq> 



The necessary condition (44) is found to be satisfied both 

 for the terms of the first and for those of the second order. 



Variable Current-density. — The preceding investigation 

 assumes the current-density to be constant at all points in the 



