34 



Proceedings of the Royal Irish Academy. 



in which we note that the subjects of volume integration vanish, and that, on 

 the surface S, c&ldn = W, and 9 = 9. 



For a sphere surrounding a source da = eVZo), where dw is an element 

 of solid angle, and 



ldd> , v m 

 - -£- da ->- - - — 

 r da 4tt7- 



n 



da^> 



m 



7' 



where r is now measured from P to the source ; also 



m 



4^ 



1(1 



dn \r 



da^~ 0. 



For a doublet 9 tends, at a distance £, to the form - (ju/47te 2 ) cos 6, where 

 is the angle which £ makes with the axis. On the sphere r 1 may be 

 replaced by r l - r~ 2 i cos 6', where >• is now measured from P to the doublet, 

 and 0' is the angle which £ makes with the direction of r. Thus 



rdn 



da-> 



M^.lcosfl'V-^ 



4- 

 2 u 



cos 0' cos 6 dw 



where \ is the angle which the axis of the doublet makes with the direction 

 of r. Also 



\+£{l)*-> 



fi cos 8 3 /f cos 0' 



47rr 



I 



47T£ 2 3e 



cos 6 cos O't/u) 



rfo- 



= - 7T — COS Y. 



Thus the limit form, for vanishing of all the t's, of equation (1) is 



f- 3 fl\ 



dS 



W 



dS 



2 — + 2 -„ cos y. 

 r r "■ 



(2) 



The left-hand side is the gravitation potential at P due to a double-sheet 

 in B of strength 9, and the right-hand side is the combined potential at P 

 due to a surface density TV in S, particles of mass - m at the sources, and 

 gravitation doublets of moment - y. at the liquid doublets. So the result is 



M*) = V (JF, - m, - y). (3) 



6. In the above it has been tacitly assumed that the region of integration 

 is finite. Modifications may be necessary if the relevant region extends to 

 infinity. 



