607] 
A MEMOIR ON PREPOTENTIALS. 
363 
69. In the case of the circumference, it is assumed that the attracted point is 
not on the circumference, k not = /; and the function under the integral sign, and 
therefore the integral itself, is in every case finite. In the case of the circle, if k 
be an interior point, then if 2q — l be =0 or positive, the element at the attracted 
point becomes infinite; but to avoid this we consider, not the potential of the whole 
circle, but the potential of the circle less an indefinitely small circle radius e having 
the attracted point for its centre; which being so, the element under the integral 
sign, and consequently the integral itself, remains finite. 
It is to be remarked that the two integrals are connected with each other; viz. 
the circle of the second integral being divided into rings by means of a system of 
circles concentric with the bounding circle x* + y s — f 2 , then the prepotential of each 
ring or annulus is determined by an integral such as the first integral; or, analytically, 
writing in the second integral x = r cos 0, y = r sin 0, and therefore dxdy = r dr dd, the 
second integral is 
viz. the integral in regard to 6 is here the same function of r, k that the first 
integral is of /, k ; and the integration in regard to r is of course to be taken 
from r = 0 to r=f But the 0-integral is not, in its original form, such a function 
of r as to render possible the integration in regard to r; and I, in fact, obtain the 
second integral by a different and in some respects a better process. 
70. Consider first the ring-integral which, writing therein as above x — f cos 0, 
y =f sin 0, and multiplying by 2 in order that the integral, instead of being taken 
from 0 to 2-77-, may be taken from 0 to nr, becomes 
Write cos|-0 = V#; then sin|6 = Vl — x, sin 6 = 2a?* (1 — x)%; dd = — x~^ (1 — x)~* dx; 
cos d — — 1 -f 2#; 0 = 0 gives x = 1, d = nr gives x — 0, and the integral is 
2 s 1 / s p#!«- 1 (1 _/p)!* -1 dx 
(/+ *) s+2? J o (T - ux)i* +< i ' ’ 
f; 
if for shortness 
The integral in x is here an integral belonging to the general form 
2«—i fs 
Ring-integral = H (H H ±s + q, u). 
46—2