172 General Analytical Theorems [OH. vn 



and hence, on substituting in equation (113) for Sp and &K, 



R 



Integrating by parts, this becomes 



sw= 



fff 2 ftK , dK , dK ,\ -, ,, j 



+ JJJ 8^ [* ^ + % + ^ *) y 



[f[(d (R z dK\. d fR* dK\s 



- Ma" a"* 1 "-*- ^ + ^~ (- T ir 8 2/ + 

 JJJ (dx\87r dr J dy\87r dr J 



or, rearranging the terms, 



rrrfr dv R*tdK\ d /R* dK\]^ r~ 



SW= \\p o- + - -5- 1 -5- Q- T -T~ ^ +... 

 JJJ ([; fa 87r\dx J dx\87r 8r/J L J 



Comparing with expression (112), we obtain 



8F 



etc., giving the body forces acting on the matter of the dielectric. 

 197. This may be written in the form 



w_ -r-^^ l^ 2 d A] 

 ~ p STT 8^ + 8^V87r T 9r 7 ' 



Thus in addition to the force of components (pX, pY, pZ) acting on the 

 charges of the dielectric, there is an additional force of components 



_^8^ _^!<^ _^^ 



8?r das ' 8-7T dy ' 8-rr dz 



arising from variations in K y and also a force of components 



l/^ 2 W\ Lf^l d A\ ^(^ ?K\ 



dx \87r T dr)' dy V87T T 8r ) ' 8* UTT T 8r J ' 



which occurs when either the intensity of the field or the structure of the 

 dielectric varies from point to point. 



