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



For we may apply the above formula (5) to all space, and then 

 the only surface integral being that due to the infinite sphere, 

 which vanishes, we have 



If however, the above conditions are fulfilled by a function V, 

 except that at certain surfaces S its first parameter is discon- 



tinuous, let us draw on each side of the surface 8 surfaces at 

 distances equal to e from 8, and exclude that portion of space 

 lying between these, which we will call Si and $ 2 . 



If the normals are drawn into r we have 



The surface integrals are to be taken over both surfaces Si and 

 $ 2 and the volume integrals over all space except the thin layer 

 between Si and $ 2 . This is the only region where there is discon- 

 tinuity, hence in r the theorem applies, and 



(i/) 



i dV rr i 



-^-dSi I 

 r on-, JJ&r 



^. vvrw*) 



s,r 3n 2 



Now let us make e infinitesimal, then the surfaces Si, S 2 

 approach each other and S . V is continuous at S, that is, is the 



same on both sides, hence, since (-) = ^ (-) , in the limit 



oni \rj dn 2 \rj 



the first two terms destroy each other. This is not so for the 

 next two, for ^ is not equal to because of the discontinuity. 



