726 
PROFESSOR CAYLEY ON PREPOTENTIALS. 
1 ~ j Csin s_1 0{(/ 2 — x 2 ) + (/ 2 + x 2 — 2x/ cos 
V J (y , 2_[_ x 2 — 2/k cos 0)i s+2 ’ 
the inferior and superior limits being here the values of 5 which correspond to the points 
N, A respectively, say $+a, and 0= 0; hence, reversing the sign and interchanging the 
two limits, the value of — jV -22 sin® -1 0 d<p is the above integral taken from 0 to a. But 
similarly the value of +J^ 1_2? sin s-1 0 is the same integral taken from a to -k ; and for 
the two terms together the value is the same integral from 0 to tt ; viz. we thus find 
Disk-integral =xn^J ^ (/> +»»-2/S,cos8)fr+« 1 
viz. writing as before cos x &c., and y, this is 
= (i — q)(x + /)•+ 2 2- 2 { ~ (x +/)* • i S 5 2 S + ^ w )+n(is, Is + g-— l)j. 
81. As a verification, suppose that z is indefinitely large : we must recur to the last 
preceding formula ; the value is thus 
. — cos0 
viz. this is 
f 
— (i _ ~ g) x s+ 2 g - 1 J ^ sin*~ — cos + [1 — (s+ 2g) cos 2 5 ] >dt 
where the integral of the first term vanishes ; the value is thus 
= (l ^y x » + 2 g f o sinS_1 P “ 0 + 2 ?) cos 2 ^] d6, 
where we may multiply by 2 and take the integral from 0 to |. Writing then 
sin fi=\/ x, the value is 
= (i 4)* s+2g J 0 1 “ O+M 1 s) K 1 “ 
1 l(«+2 g) \ _ r^ri \-g 
J’-ms+i)' i*+¥ 
and hence the value is 
_/ 8+I rigTj 
viz. this is=^yj*y 5_1 dxdy , over the circle x 2 -\-if=f 2 , as is easily verified. 
82. Reverting to the interior point z < /', 
1 1 4 - / 
where the integral is— ( 
