PROFESSOR CATLET ON PBEPOTENTI ALS . 
685 
order that this may be so, then the formula determines the distribution, viz. it gives the 
value of g, the density at a point (x , . . . z, w) of the surface. 
13. In the case where the surface is the plane w= 0, viz. in the case of the potential- 
plane integral, 
ydx . . . dz 
{(« — a?) 2 . . . + (c— ar^ + e 9 }** i 
(B) 
(e assumed to be positive) ; then, since every thing is symmetrical on the two sides of the 
plane, V' and Y" are the same functions of (a .. . c, e), say they are each=V ; W', W" 
are each of them the same function, say they are each = W, of (x ... z, e) that V is of 
(a . . . c, e), and the distribution-formula becomes 
_ rg s -i) /dw\ 
§ 2(ri) s+1 V de ) e J 
(B) 
viz. this is also what one of the prepotential-plane formulae becomes on writing 
therein 
q= 0, or Negative. — Nos. 14 to 18. 
14. Consider the case ^=0. The prepotential of the disk is 
f-w (logE+N_logs "- ); 
and to get rid of the constant term we must consider the derived function in regard to 
s, viz. this is 
2(r*)« i 
r ^ •*> 
and we have thus for the whole surface 
dV _ 2(Tj)« l 
d* u 2 T%s 
where Q, which relates to the remaining portion of the surface, is finite ; we have thence, 
writing, as before, W in place of V, 
dW 
da z 
2(r*)« 
'2 TU ’ 
or say 
T$s ( dW\ 
- 2(ri) s ' 
ds . 
15. Consider the case q negative, but —q<\- The prepotential of the disk is here 
2 fro (R-22 , . risrw ) 
and to get rid of the first term we must consider the derived function in regard to a, 
viz. this is 
2(ri)T( g +l) . 
* 2 r(**+j) ’ 
4 y 2 
