82] 



NEWTONIAN POTENTIAL FUNCTION. 



161 



and making R 2 R = e, lim J^ = lim R^= R, 



47TO- 



and on the outside for h = E, 



dV 

 dh 



47TCT. 



Within we have everywhere 



dV 



V= const., -rr = 0. 

 dh 



Hence 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 47nr. 





FIG. 39. 



Consider now any surface distribution of surface density a. 

 Apply Gauss's theorem to a small tube of force bounded by portions 

 of two equipotential surfaces d^ and c2 2 on opposite sides of the 

 element of the attracting surface dS (Fig. 39). The flux out from 

 the tubes is 



and this must be equal to 4- 4?r times the matter contained in 

 the tube, which is adS. Therefore 



But if the length and diameter of the tube are infinitesimal 

 and cE 2 are the projections of dS, 



where n is the normal to the attracting surface. Accordingly 

 F 2 cos (Fn) dS - F t cos (Fn) dS = 



W. E. 



11 



