PROFESSOR CAYLEY ON PREPOTENTIALS. 
725 
Bat we have 
and thence 
that is 
/sin 0 fcosd—x 
sin cos<p= , 
/ sin 9 2 7 /(/— x cos'd) d& 
tan <p = -/ — i — , sec 2 <p d<p =-¥7 7 — 4 2 ; 
r /cos0 — x’ r r (/cosfl — xy ’ 
d<p = 
f(f—x cos 
or, what is the same thing, 
f{f—x cos Q)dO 
- / 2 + x 2 — 2x/cos 0’ 
M(/ 2 -^) + (/ 2 + x 2 -2x/cosfl)} . 
/-fx 2 — 2x/cos 0 
^/ s_1 (*”■ sin s_I 0{(/ 2 — x 2 ) + (/ 2 + x 2 -2x/cos ( 
{/ 2 + x 2 — 2x/cos 0}i s+ 2 
ill!? r ^ r i 
i— 1 
and the expression for the disk-integral is therefore 
i-srJ. 
79. Writing as before cos x, sin See., and this is 
= ( i — 5 ) ( x +/)* + 2 «- 2 {^+/) 2 n ^ 5 ’ + W ) + n ( i S ’ a s > a 5 + 2 '— 1 , M )| — iZTg r (| s +|)* 
As a verification observe that if z=0, each of the Il-functions becomes 
= | (1 — ®) 2 dx , — ; 
2 , 2 s-2 z 1 - 2 ? risris . f i ~ 2 i rj-sri 
hence the whole first term is= --- . — p s — , viz. this is=^— — rrx 2 anc ^ 
complete value is 
■~<1 
i-q r(is + |) 
l ru ,, ri 
x 2 & 1 2 j f\-2 a 5 l-2 2 ) 
~i-g 
vanishing, as it should do, if/=e. 
80. In the case of an exterior point %>f the process is somewhat different, but the 
M 1 
result is of a like form. We have 
Disk-integral = (g! 1-22 — g 1 - 2 a )sin* -1 p d<p, 
gi referring to the point M' and to the point M. Attending first to the integral 
jy ~ 22 sin 4-1 <p d<p, and writing as before/cos 6 =z+§ cos cp,f sin Q=g sin <p, this is 
r s _ 1 r sin*-^ d<p 
_ ■ ' J I X 2 +/2 - 2 xf cos 0 }* s+2 
5 d 2 
