Ellipsoidal Shells and of Solid Ellipsoids. A09 



which is to be interpreted as the total step of potential 

 involved in carrying the matter, supposed initially in a thin 

 homoeoid (mass m) of which the equation is %{x 2 j(a 2 -^u)\=k 

 from the level surface Xx 2 /a 2 = k to infinity in the manner 

 described. Thus matter m distributed in a homoeoid on the 

 latter surface has the same potential at the surface and at all 

 external points as the same matter had when in the original 

 homoeoid. This is for confocal bomoeoids what Maclaurin's 

 theorem is for solid ellipsoids, and indeed Maclaurnr's theorem 

 flows from it at once since a solid ellipsoid may be supposed 

 built up of a succession of homoeoids. It is to be observed, 

 however, that this theorem of equivalence of homoeoids on 

 confocal surfaces is only a particular case of Green's very 

 general theorem of equivalence. Maclaurin's theorem and 

 the analogous theorems for shells have been explained in 

 § 22. and it is not necessary to pursue the subject here. In 

 § 18 the extension for a shell to a solid ellipsoid has also been 

 fully discussed. 



31. The potential of a thin homoeoid at a point on its 

 surface or anywhere in the interior is given in (12) . From 

 this we can obtain the potential of a thick homoeoid at a point 

 within the interior hollow. TTe shall suppose that the 

 equations of the outer and inner surfaces are respectively 



.2 2 



2 -= = 1 , and 2 *- 5 = Ji . 

 a- a- 



We have therefore only to integrate (12) with regard to h 

 from Jc = h to k = l. Thus for the thick homoeoid 



Y=wpabc(l-h) r — -- du - — — . (U) 



Jo \/(a 2 + u){0 2 + u)(c 2 + u) K 



The potential produced by this homoeoid at an external 

 point may be found as follows : The squares of the semi-axes 

 of the interior ellipsoidal hollow area 2 /*, b 2 h, c 2 h, where h < 1 . 

 The potential Y / at the external point t\ g, li (here h is a co- 

 ordinate), due to an ellipsoid of density p filling the hollow i^ 



(45) 



, M (1—2— — )du l 



\ Jf 173 \ a-li + u^/ 



\/(a 2 li + u^ttfii -f u { )(c 2 h + u x ) 

 where A x is the positive root of the equation 



cr/t +?/i 



