607] 
A MEMOIR ON PREPOTENTIALS. 
329 
to get rid of the constant term we must consider the derived function in regard to 8, 
viz. this is 
2(T£) S 1 
~P 
and we have thus for the whole surface 
dV 
da 
-o »(rjyi 
V P ni „ > 
r±8 
where Q, which relates to the remaining portion of the surface, is finite; we have thence, 
writing, as before, W in place of V, 
da 
~P 
say 
P = 
dW_ 2 (r|) s 
ri* ’ 
dWn 
r $8 
2(r hY 
da ) a= 
15. Consider the case q negative, but — q<^. The prepotential of the disk is here 
'«(Tirp«. iyr g 
P rjs i-2 9 + 2 r(J« + o) 
+ ... 
to get rid of the first term we must consider the derived function in regard to a, 
viz. this is 
2(I+)T(g+l). 
9 r(i»+i). ’ 
whence, for the potential of the whole surface, 
dV Q 2(ryr( g + l) 
* W P r (i* + i) ’ 
where Q, the part relating to the remaining portion of the surface, is finite. Multiplying 
by a 2( i +1 (where the index 2^ + 1 is positive), the term in Q disappears; and writing, 
as before, W in place of V, this is 
or, say 
,29+1 
p = - 
dW_ 2(r±yr(q+l) 
da 
= ~P 
r^S + q 
T (|s + q) 
2 (r|) s r (q 4-1) 
.29+1 
dW' 
da 
8 = 0 
viz. we thus see that the formula (A*) originally obtained for the case q positive 
extends to the case q — 0, and q = — but — q < £; but, as already seen, it does not 
extend to the limiting case q — — 
16. If q be negative and between —and —1, we have in like manner a formula 
dV O 2( r ^ r ( g + !) 
da ^ P T ($s + q) 
C. IX. 
42