682 
PROFESSOR CAYLEY ON PREPOTENTIALS. 
and 
r (fr+g) 
*(T*)T(a+i) 
e=0 
(A*) 
which will be used in the sequel. 
It will be remembered that in the preceding investigation it has been assumed that 
q is positive, the limiting case q=0 being excluded ■j*. 
q——\. — Nos. 8 to 13. 
8. I pass to the case q-=.—\, viz. we here have the potential-surface integral 
gdS 
{{a—xf ... + (c— 2) 2 - f (e—w)’ 2 } is 
(C) 
it will be seen that the results present themselves under a remarkably different form. 
The potential of the disk is, as before, 
2 ( r i) s T r*~ l dr 
e rj s J(r s +s s )s-*’ 
where § here denotes the density at the point N ; and the value of the r-integral 
-d/t . , . s 2 s 4 \ r*»r± 
=R(l+term 5 in ^ ^ . . .) 
Observe that this is indefinitely small, and remains so for a point P on the surface ; 
the potential of the remaining portion of the surface (for a point P near to or on the 
surface) is finite, that is, neither indefinitely large nor indefinitely small, and it varies 
continuously as the attracted point passes through the disk (or aperture in the material 
surface now under consideration) ; hence the potential of the whole surface is finite for 
an attracted point P on the surface, and it varies continuously as P passes through the 
surface. 
It will be noticed that there is in this case a term in V independent of a ; and it is on 
this account necessary, instead of the potential, to consider its derived function in 
regard to a; viz. neglecting the indefinitely small terms which contain powers of 
a T 
•g, 1 write 
(tV __ 2 (ri)« +1 
da r(is-i) ?■ 
The corresponding term arising from the potential of the other portion of the sur- 
face, viz. the derived function of the potential in regard to a, is not indefinitely small ; 
and calling it Q, the formula for the whole surface becomes 
dV _ 2(ri) s+1 
t This is, as regards q, the case throughout ; a limiting value, if not expressly stated to he included, is 
always excluded. 
