127] 



SURFACE DISTRIBUTIONS. 



369 



Thus there is in like manner a discontinuity in the first derivative 

 of the potential in the direction 

 of the normal, on passing through 

 the attracting surface, of the 

 amount 4#<?. 



Consider now any surface 

 distribution of surface density 0. 

 Apply Gauss's theorem to a small 

 tube offeree bounded by portions 

 of two equipotential surfaces d^ 

 on opposite sides of and 



Fig. 133. 



times the matter contained in the 



near to the element of the attrac- 

 ting surface dS (Fig. 133). The 

 flux out from the tubes is 



and this must be equal to 

 tube, which is 6dS. Therefore 



But if the length and diameter of the tube are infinitesimal 

 and dZ! 2 are the projections of dS, 



1 n) } d 2J 2 = d S cos (F 2 n) 

 where n is the normal to the attracting surface. Accordingly 



F 2 cos (F 2 n) dS F cos (F^ri) dS = 

 and since 



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

 then that in general, on traversing a surface distribution, the normal 

 force has a discontinuity equal to 4j> yt 6. 



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

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



94) 



or 



l cos t 



WEBSTER, Dynamics. 



dV , dV 



K h o = 



r 



cos 



24 



