A MEMOIR ON PREPOTENTIALS. 
359 
607] 
[x 2 + ... + z 2 4- (« 
w y*\h«+ q ’ 
63. It has been tacitly assumed that 4- q is positive; but the formulae hold 
good if + q is = 0 or negative. Suppose + q is 0 or a negative integer, then 
r (^s 4- q) = oo, and the special term involving e~ 2q or e~ 2q log e vanishes; in fact, in 
this case the r-integral is 
+ dr, 
where (r 2 4- e 2 )~& s+q) has for its value a finite series, and the integral is therefore equal 
to a finite series A 4- Be 2 + Ce 4 + &c. If 4- q be fractional, then the T of the negative 
quantity + q must be understood as above, or, what is the same thing, we may, 
instead of F (-|s 4- q), write 
(Eli! . 
sin (%s + q) 7r T (1 — q — |s) ’ 
thus, q being integral, the exceptional term is 
- (-\q P -nrjssin(jg + g)Tr.r(l-g-fo) , R 
K ’ (H) 2 r (1 - q) g « • 
For instance, s = 1, q = —2, the term is 
sin (— f 
(ny.rs 
7r) Ti , 
Mog 
R 
or, since r| = |.^r^, and T3 = 2, the term is +fe 4 log—, agreeing with a preceding 
result. 
Annex III. Prepotentials of Uniform Spherical Shell and Solid Sphere. 
Art. Nos. 64 to 92. 
64. The prepotentials in question depend ultimately upon two integrals, which 
also arise, as will presently appear, from prepotential problems in two-dimensional space, 
and which are for convenience termed the ring-integral and the disk-integral respect 
ively. The analytical investigation in regard to these, depending as it does on a 
transformation of a function allied with the hypergeometric series, is I think interesting. 
65. 
This is 
Consider first the prepotential of a uniform (s 4-1 )-dimensional spherical shell. 
dS 
{(a — x) 2 +... 4 (c — z) 2 + (e — w) 2 \^ +q ’ 
the equation of the surface being x 2 4- ... + z 2 4 w 2 —f 2 ; and there are the two cases 
of an internal point, a? + ... 4- c 2 4- e 2 <f 2 , and an external point, a 2 + ...+c 2 +e 2 > f 2 . 
The value is a function of a 2 + ... + c 2 + e 2 , say this is = k 2 . Taking the axes so 
that the coordinates of the attracted point are (0, 0, k), the integral is 
dS