kp Ca< 



454 Mr. It. Hargreaves cm 



the area of the section of the ellipsoid by the plane through 

 xyz, and <r that by the parallel plane through the centre, 

 <r=ff- (l— OTi 2 /^/) , and o" = ?r/ Vu K . Using these values in 

 (129) we have 



? (130) 



In the isotropic problem k (which is A p in the general 

 case and 1 — 2jo 2 in the motional case) is 1, and also u v which 

 is Xpl 2 + 2p'mn becomes 2/ 2 or 1. Hence 



*=&£*•=£'' •• • • • (131) 



■cr being the mean area of all sections of the ellipsoid drawn 

 through the point whose potential is considered. At the 



centre the potential is ~~ 9, and the potential of a conductor 



Z7T 



with charge pr n is p ~^ The expression for the energy is 

 P e * o or e ^o 



57T 57TT 



with volume-distribution ; for a conductor 6 replaces 5. [On 



the general^ validity of (130) vide infra.'] ^ 



In working by differential methods the use of p = ^-p 



for density, and J2XX' + J :£««' for energy has been followed. 



The older method has 1tt/? and -i (XXX' + 2««') ; the 



values in that notation are got by multiplying the expressions 

 for potential and also for energy by Air. The older fashion 

 gives here the simpler formulae, and they stand thus: — 



-\jri=2p Q a, with 2p o- Q at centre, and -~<r for a conductor I 

 Energy ^/Vo^o or ^"•°"° for vol. case, -f- . ^ for conductor 1 



D T '0 T J 



For the general seolotropic case the area of each element 

 is modified by the multiplier kju p or A p ju p depending on the 

 exposure of the plane in relation to the aeolotropic axes. For 

 the motional case the multiplier is 



(l-2^ 2 )/(l-U 2 /V 2 ) or J— g?V - 1 .+ - 1 \ 

 r /V ; ; 2 Vl-U/V + 1 + U/V/' 



U being lu + mv + nw ; i.e., apparently it is the mean of 



