120, 121] POTENTIAL OF CONTINUOUS DISTRIBUTION. 355 



Since r and r 2 are constant, 



' 



Now since / / / dm = M, the whole mass of the body K, the 



K 



above is 

 54) f<F<f. 



Accordingly for an external point V is finite. 



If E is the distance of x,y,z from some point in or at a finite 



distance from K, 



c wv EM w r EM 



oo) <E V< 



If now we move off x, y, a to an infinite distance we have 

 lim = lim = 1 



E = oo r 2 B = r l 



and accordingly since EV lies between two quantities having the 

 same limit, 



56) lim(JRF) = Jf. 



.R=ao 



We say that V vanishes to the first order as E becomes infinite. 



121. Derivatives. Consider the partial derivatives of V by 



x, ij, e. 



The element dm at a, &, c produces the potential 



dV=^ 

 at x,y,e. 



Differentiating by x, (dm and a, ~b, c being constant), we have 



By 118,45)^ = ^ 



58) ^ 



Now 



p' /-v\ & - (& / \ 



59) -^ r - = cos(rx), 



where the direction of r is taken from a, &, c to x,y,z. This being 

 the derivative for that part of the potential due to dm, we have to 



23* 



