﻿DEKIVATIOX OF POISSOX^S . EQUATION B7 



By symmetry, 



jJxdS =/fijdS =f/zdS =/fxydS ^^^ffxzdS ^ ffyzds = (\ 



and 



iJifdS =JJz'dS =jfx'dS. 

 By calculation, we find 



ffx'dS i-R\ 



O 



Let V be the average value of Y on the sphere, then we have on 

 substituting in (1) . 



4.fl^(F-F„)=f.fin^lI 41 „-■... (^) 



^ ox oy oz 



where the terms that are omitted contain powers of F higher than 

 the fourth. 



To calculate the quantity V - Y,,, let l^^ be the potential at 0 

 due to the matter outside the sphere, and Y the potential due to 

 the matter within. As the sphere becomes smaller the density ap>- 

 proaches its value /'o at the center, if the density is continuous. 

 Therefore, 



R 



Fo Fi ^ Y. = Ti + / 4^Pordr= Fi + 2^PoB\ 



and 



Y= (average value on the surface due to the matter without) 

 - (average value on the surface due to the matter within). Or, 

 by applying Gauss's theorem of the arithmetic mean, 



Y= (potential at 0 due to matter without the sphere) + (po 

 tential on sphere if all the matter within were concentrated at the 

 renter) 



Y^ + j^PoR'. 



Hence 



4 



9 



F - Fo = - ^PoR' - 2^PoR" = - ~^PoR" 



Sul^stituting in (2), we have 



, a-F d- F , aM' 



" .Ti^^ ^ -:tE' ( — - ] + terms above R\ 



■> dx- dif- dz' 



