714 
PEOFESSOE CAYLEY ON PEEPOTENTIALS. 
where (r 2 +e 2 ) -cis+s) has for its value a finite series, and the integral is therefore equal to 
a finite series A+Be 2 +Ce 4 +&c. If \s-\-q be fractional, then the T of the negative 
quantity l-s-j-# must be understood as above, or, what is the same thing, we may, instead 
(r 1 ) 2 
of r(-|s-l-g'), write s - n 5 thus, q being integral, the exceptional term is 
_ r v, -a, rja sin (fr + g)?r • f(l —q — ^s ) , R 
— t—) e (T^T{l-q) i0 & e ’ 
for instance, s=l, q=— 2, the term is 
or, since r§=f . ^ 
result. 
, sin ( — hr) Tf . R 
^~ ri)».ra Io g7- 
ri, and T3=2, the term is -f-fe 4 log^, agreeing with a preceding 
Annex III. Prepotentials of Uniform Spherical Shell and Solid Sphere . — 
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 respectively. 
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. Consider first the prepotential of a uniform (s+1) dimensional spherical shell. 
This is 
(a — a ?) 2 . . 
dS 
+ (c — z) 2 + (e— M>) 2 p s+S ’ 
the equation of the surface being x 2 . . . +z 2 +w 2 =/ 2 ; and there are the two cases of an 
internal point, a 2 .. . -\-c 2 -\-e 2 <f 2 , and an external point, a 2 . . . -\-c 2 -\-e 2 >f 2 . 
The value is a function of a 2 . . . -fi c 2 -\-e 2 , say this is =z 2 ; and taking the axes so that 
the coordinates of the attracted point are (0 ... 0, z), the integral is 
d S 
^ 2 ...+* 2 +(x-w) 2 p +2 ’ 
where the equation of the surface is still x 2 . . . -{-z 2 -\-w 2 =f 2 . Writing x=f% . . . z=j%, 
w=fco, where | 2 . . . -\-% 2 -\-a 2 =l, we have dS = or the integral is 
>(/ 2 -2x/«> + x 2 )i*+2- 
Assume %=px , . . . %=pz, where x 2 . . . -\-z 2 = 1 ; then p 2 -\-a 2 =l. Moreover, d \ . . . d£, 
—p s ~ l dp d%, where d% is the element of surface of the s-dimensional unit-sphere 
x 2 . . . -\-z 2 — 1 ; or for p, substituting its value \/l — a 2 , we have dp = > anc ^ thence 
