162 THEORY OF NEWTONIAN FORCES. [PT. I. CH. IV. 



and since F 2 cos (Fri) = ( ) , 



\dn/ 



F\ /3F\ 



-5 = 



/1/1 \9r&/2 



6 . 



The normal to 8 is here drawn toward the side 2. We find 

 then that in general, on traversing a repelling surface distribution, 

 the normal force has a discontinuity equal to 4?ro-. 



This is Poisson's equation for a surface distribution. If we 

 draw the normal away from the surface on each side, we may 

 write 



r ^ pT~ = ~ ^Tro-, 



or Fj, cos (Fin,) + F, cos (jP 2 w,) = P 1M + F^ = 4>7ro: 



83. Greenes formulae. Let us apply Green's theorem to 

 two functions, of which one, F, is the potential function due to 



any distribution of matter, and the other, U = - , where r is the 



r 



distance from a fixed point P, lying in the space r over which we 

 take the integral. Let the space r concerned be that bounded by 

 a closed surface S, a small sphere 2 of radius e about P, and, if P 

 is wihsttt 8, a sphere of infinite radius with center P. 



FIG. 40. 



Now the theorem was stated in 33 (2) for the normal drawn 

 in toward r, which means outward from S and S, and inward 

 from the infinite sphere, as 



and since 



