PROFESSOR CAYLET ON PREPOTENTIALS. 
717 
68. The integrals referred to as the ring-integral and the disk-integral arise also from 
the following integrals in two-dimensional space, viz. these are 
C yt-'dS C y s - l dxdy 
J + J {{x-xF + y*}***’ 
in the first of which dS denotes an element of arc of the circle x 2 -\-y 2 =f' 2 , the integra- 
tion being extended over the whole circumference, and in the second the integration 
extends over the circle x 2 -\-y 2 =f 2 ; y*~ l is written for shortness instead of viz. 
this is considered as always positive, whether y is positive or negative ; it is moreover 
assumed that s — 1 is zero or positive. 
Writing in the first integral x=f cos 6, y~f sin Q, the value is 
» f I ^ 
—J J (/ 2 — 2 x/ cos 9 + * 2 )i s+2 ’ 
viz. this represents the prepotential of the circumference of the circle, density varying as 
(sintf) 8-1 , in regard to a point x=x, y = 0 in the plane of the circle; and similarly the 
second integral represents the prepotential of the circular disk, density of the element at 
the point (x, y)=y s ~ l , in regard to the same point x=x, y— 0, it being in each case assumed 
that the prepotential of an element of mass gd& upon a point at distance d is = J^. 
69. In the case of the circumference, it is assumed that the attracted point is not on 
the circumference, z not =f; and the function under the integral sign, and therefore the 
integral itself, is in every case finite. In the case of the circle, if z be an interior point, 
then if 2^—1 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 s 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 in rings by means of a system of circles con- 
centric with the bounding circle A’ 2 -f-?/ 2 ==/’ 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 d, y=r sin 0, and therefore dxdy=rdrdQ, the second integral is 
(sin d$ 
+ k 2 — 2 xr cos 
0)^+2’ 
viz. the integral in regard to 6 is here the same function of r, z that the first integral is 
of/, z ; and the integration in regard to r is of course to be taken from r— 0 to r=f. 
But the ^-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, 
5 c 2 
