350
A MEMOIR ON PREPOTENTIALS.
[607
Cl,2
internal point 7^ 1
h 2
; as e approaches zero the positive root of the original
equation 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 integration 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 + l)-coordinal sphere x 2 + ...+z 2 +w 2 = f 2 , and
ellipsoid %, + •" + iL+-L =: 1» or sa y t° spherical and ellipsoidal shells; but I have
j - h- k-
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 p, or distribution over the spherical surface, which shall produce these potentials
respectively. The result (see Annex VI.) is
f _dS
J {(a — x) 2 + ... + (c — z) 2 + (e — w) 2 p'~*
over the surface of the (s + l)-coordinal sphere x 2 + ... + z 2 + w 2 =/ 2 ,
2 (IM) S+1 f* 1
= .A .. —A for exterior point ic > f
and
2 (rA) s+1 / s 1 „ . . . . . ,
= rT7i V for in f enor POmf K </>
r (is + i) f 1
where tc 2 = a 2 + ... + c 2 + e 2 . Observe that for the interior point the potential is a mere
constant multiple of f
The same Annex VI. contains the case of the s-coordinal cylinder x 2 + ... + z 2 — f 2 ,
which is peculiar in that the cylinder is not a finite closed surface ; but the theorem
C is found to extend to it.
49. As regards theorem D, we might in like manner obtain potentials relating
to the (s + l)-coordinal sphere x 2 + ... + z 2 + w 2 = f 2 and
... .. x 2 z 2 w 2 ,
ellipsoid -+...+^ + -=1;
but I confine myself to the case of the sphere (see Annex VII.). We here assume
values V' and V" belonging to an internal and an external point respectively, and
thence obtain a value p, or distribution over the whole (s + l)-dimensional space,
which density is found to be =0 for points outside the sphere. The result obtained is
V=
dx ...dz dw
{(a — x) 2 + ... + (c — z) 2 + (e — w) 2 }* s *
