686 
PROFESSOR CAYLEY OX PEEPOTEXTIALS. 
whence for the potential of the whole surface 
=Q— S' 
2(ri) s r(?+i) 
? r(is-ts) 5 
where Q, the part relating to the remaining portion of the surface, is finite. Multiplying 
by s 2?+1 (where the index 2y+l is positive), the term in Q disappears ; and writing, as 
before, W in place of V, this is 
# da - 
2(ri)*r(< ? +i) 
r \s + q ’ 
or say 
r(fr+g) 
2(T*)T(g+l) 
( s ' 5 
dW\ 
d* . 
viz. we thus see that the formula (A*) originally obtained for the case q positive 
extends to the case q=0, and q= — , but —q<j>; hut, as already seen, it does not 
extend to the limiting case q= 
16. If q be negative and between — ^ and —1, we have in like manner a formula 
dV 
da 
=Q— « 
2(Tl)T( g +l) _ 2? _ 1 . 
r(t»+*) 
but here 2g'+l being negative, the terms 22 ’Q does not disappear: 
to be treated in the same way as for q=—^, and we arrive at 
dW , 
■w+* 
Il2q+1 
dW" ) 
da" ) 
4(T±yT(q + l) 
r(is+ q ) i e; 
the formula has 
viz. the formula is of the same form as for the potential case q = — 
formula does not hold good in the limiting case q— — \. 
17. We have, in fact, here the potential of the disk 
whence 
2 (rip (R 2 9 . ri? ) 
— r(i«) f j 2 “* lo S 8 r(i*-i)/ ; 
w 2{T^y 
da~Q r(i S -l) g( 2gl °g*)i 
Observe that the 
since in the complete differential coefficient a + 2s log s the term s vanishes in compari- 
son with 2s log s ; and then, proceeding as before, we find 
i dw 1 i dW" -8(r±y 
s' log s' da 1 a" log a" da" T (^s — 1 ) ^ ’ 
but I have not particularly examined this formula. 
18. If q be negative and > — 1 (that is, —§'>1), then the prepotential for the 
disk is 
_ (rj)‘/R-»g , ±s + q R ~ 2g ~ 2 
~ ? T-gS y~2q' 1 -2q-2 
and it would seem that in order to obtain a result it would be necessary to proceed to 
a derived function higher than the first ; but I have not examined the case. 
