PEOFESSOE CATLET ON PEEPOTENTIALS. 
705 
47. We may in the foregoing results write e=0 ; the results, writing therein s + 1 for 
s, and in the new forms taking (a . . . c, e) and (x . . . z, w) for the two sets of coordinates 
respectively, also writing q—\ for q, would give integrals of the form 
P pdx...dzdw 
J {{a—xf.. . + (c~^) 2 +(e— w) 2 p s+2 
for the (s-j-l)coordinal sphere and ellipsoid x 2 . . . + z 2 -\-w 2 =f 2 and J 2 • • • + f 2+^2 = l>* 
say these are prepotential solid integrals; and then, writing q=—\, we should obtain 
potential solid integrals, such as are also given by the theorem D. The change can be 
made if necessary ; but it is more convenient to retain the results in their original 
forms, as relating to the s-coordinal sphere and ellipsoid. 
There are two cases, according as the attracted point (a . . , c) is external or internal. 
.X 2 
For the sphere: — For an external point > / 2 ; writing e — 0, the equation ^^=1 
has a positive root, viz. this is ; and 0 will have, or it maybe replaced by, this 
value y 2 —f 2 : for an internal point n 2 <f 2 ; as e approaches zero, the positive root of the 
original equation gradually diminishes and becomes ultimately =0, viz. in the formulae 
0 is to be replaced by this value 0. 
« 2 c 2 . . 
For the ellipsoid: — For an external pointy. . . +^>1; writing e=0, the equation 
Ci~ (& ' ' ... 
. . . + fl_j_^ 2 =:l has a positive root, and 6 will denote this positive root: for an 
internal point p . . . < 1 ; as e approaches zero the positive root of the original equa- 
tion gradually diminishes and becomes ultimately =0, viz. in the formulae 6 is to be 
replaced by this value 0. 
The resulting formulae for the sphere x 2 . . . -\-z 2 =f 2 may be compared with formulae 
for the spherical shell, Annex VI., and each set with formulae obtained by direct inte- 
gration in Annex III. 
We may in any of the formulae write q=—^, and so obtain examples of theorem B. 
48. As regards theorem C, we might in like manner obtain examples of potentials 
relating to the surfaces of the (s-J-l)coordinal sphere x 2 . . . -J -z 2 -\-w 2 =f 2 , and ellipsoid 
00 ^ 
7 s * ••+*«+ F =1 ’ or say to spherical and ellipsoidal shells ; but I have confined myself 
to the sphere. We have to assume values V' and V" belonging to the cases of an 
internal and an external point respectively, and thence to obtain a value g, or distribu- 
tion over the spherical surface, which shall produce these potentials respectively. The 
result (see Annex VI.) is 
