460 Intelligence and Miscellaneous Articles. 



The densities being h, let the potential of the layer on the sur- 

 face C, in points within C, be represented by P i} and the potential 

 of the other layers by H a ; then is 



and 



dYj _ dPj , cTR t 



dn dn dn 



cTVg _ dP a dR a 



dN dN dN' 



Manifestly, for points of C, 



dK i __dR am 



dn dN '' 



and according to a known property —j 1 - + -j^- =—^eh, where € 

 represents a constant. Hence we find 



an dN 

 and in precisely the same way 



£**-'-■* 



Therefore, by the summation of both members of equations (2) and 

 (3), we get 



Lastly, I remark that this general equation is to be regarded as 

 an extension of a property given by G-auss. Let V represent the 

 potential of a system of masses m l5 m 2 , &c. situated in the points 

 Pv Pv & c "> anc ^ ^ ^he potential of masses m v m 2 , &c. in the points 

 j) l5 p 2 , &c. ; further, Y v Y 2 , &c. the values of V in the latter points, 

 and $$ v 35 2 , &c. the values of SB in the points p lf p 2i &c. ; then, 

 according to Grauss, 



SVm=253m. 



This equation is identical, since both members represent aggre- 

 gates of the same combinations. That the theorem still remains 

 valid when first the masses m are spread out upon a surface and 

 afterwards the masses m upon the same surface, is not completely 

 demonstrated by Grauss *, but " the way in which this extension of 

 the theorem can be rigorously justified with respect to its principal 

 moments only " pointed out. 



In the foregoing this theorem is demonstrated for an arbitrary 

 number of surfaces as a simple deduction from Green's equation. — 

 "Wiedemann's Anncden, 1880, No. 5, x. pp. 154-158. 



* Gauss, Allgemeine Lehrsatze in Beziehung auf die im verkehrten Ver- 

 haltnisse des Quadrats der Enlfernung wirkenden Krafte, § 19. 



