410 Prof. A. Gray on the Attraction of 



If in (45) we write uh for u^ we get 



/ ■ f 2 \ 

 T = 7rpabc —, \ == £ + u^__ t . (46) 



where V =\ 1 /h. The potential at the external point /, </, A 

 due to the complete ellipsoid is given by (20) [§ 18]. Sub- 

 tracting (46) from (20) we obtain for the thick homoeoid 



= 7rpabe \ 



(l-2-/J-V M 



V a Zj r u) 



\f(a 2 + u)(l) 2 + u){c 2 + u) 



When the internal hollow is contracted to zero, that is when 

 A=0, V becomes go, and the second integral on the right 

 vanishes. The equation then coincides with (20) as it ought 

 to do. 



32. We can now find the potential produced by a thick 

 focaloid at an external or internal point. First for an external 

 point : let two confocal ellipsoids have the same density, and 

 let the shell between their surfaces be the focaloid to be con- 

 sidered. The potentials which they produce at a point external 

 to both are in the ratio of the masses of the ellipsoids. For 

 the potential of that which has ^(x 2 ja 2 ) — \ for the equation 

 of its surface the equation is (20) [§ 18 above], and for the 

 potential of the other which has, let us say, %{x 2 j(a 2 — s)\ = l 

 for the equation of surface, the equation is the same with as 

 multiplier of the integral irp\/(a 2 — s){b 2 — s)(c 2 — s) instead 

 of irpabc. Thus for the potential of the focaloid at an external 

 point /, g, A we obtain 



\ a 2 +u/ 



\Z(a 2 +u)(b 2 + u)(c 2 +u) 



\du 



diere mf is put for the mass of the focaloid, that is 



3 irp \ abc —\/{a 2 — s)(b 2 — s) (c 2 — s) \ . 



.4 



This equation is precisely the same as (20) which gives the 

 potential at an external point for a solid ellipsoid : and just 

 as in the case of ellipsoids it follows that : — any two uniform 

 focaloids which are confocal with the same ellipsoid produce, 

 at any point external to both, potentials which are in the 

 ratio of the masses of the focaloids. 



