724 
PBOFESSOR CAYLEY ON PREPOTENTIALS. 
77. We come now to the disk-integral, 
£ y s ~ 1 dxdy 
J {(*-*)* + y»}** + *’ 
over the circle x 2 -\-y 2 =f 2 . Writing x=z+g cos<p,y=g sin <p, we have dxdy=% dg d<p, 
and the integral therefore is 
J sin® -1 <p dg dQ 
— ^ ’ 
where the integration in regard to g is performed at once, viz. the integral is 
=rzyV 2? ) sins_i $ ^ ’ 
or multiplying by 2, in order that the integration may be taken only over the semicircle, 
y= positive, this is 
=fz~ q J (? 1-22 ) sin * _1 Q 
the term (§'~ 2i ) being taken between the proper limits. 
78. Consider first an interior point »<f. As already mentioned, we exclude an 
indefinitely small circle radius s, and the limits for q are from § ==s to g>=its value at the 
circumference; viz. if here x—f cos#, y=fsxn.O, then we have /"cos Q=z-\-q cos <p, 
f sin 0=q sin <p, and consequently 
cos 0, 
• * / ■ A / Sin ^ 
sin <p =- sm 6 , = 
? \/x 2 +/ 2 — 2xf cos I 
1 / /«-* sin"- 1 6 
\{x 2 +/ 2 -2x/cosflp s+? - 1 
As regards the second term, this is = — sin® -1 <p (ftp, <p=0 to <p=?r, or, what is the 
same thing, we may multiply by 2 and take the integral from <p=0 to <p = ^. Writing- 
then sin <p = s/ x, and consequently sin s_1 <p (ftp = |# s_1 (l — x) -i dx, the term is 
gl-2 q 
— ~TZTq r(|g 4 -lp an ^ the va l ue °f the disk-integral is 
_f s ~ l C sin 5-1 6 d<p e 1 - 2 ? 
_ i — ( 1J (* 9 +/ 2 — 2x/cosS)* s+ 2- 1 | — q F(|s-f£) - 
