607] 
A MEMOIR ON PREPOTENTIALS. 
function under the integral sign is to be replaced by zero whenever the values u, t 
CV^ c 2 6 2 
are such that u is less than - + ... 4- -y-—- + - , viz. when the values belong to a 
/ 2 +1 hr+ t t 
V = 
point in the shaded portion of the strip; the integral is therefore to be extended 
only over the unshaded portion of the strip; viz. the value is 
r<-g) rfe + g)// du dt ■ r4 "' f (i + - / ' 1) - • ■ (t + (“ -fTt - ■'" - WTi-1) q 1 *"> 
the double integral being taken over the unshaded portion of the strip; or, what is the 
qq2 g2 
same thing, the integral in regard to u is to be taken from u —— + ... + ATT , + 7 
T Z ¡1 “7” z z 
(say from u = a) to u — 1, and then the integral in regard to t is to be taken from 
t — 6 to t= oo, where, as before, 6 is the positive root of the equation cr = 1, that 
„ a 2 c 2 e 2 n 
1B ’ oi /iT0 + - + h?+0 + e~ 1 - 
162. Write u = a + (1 — a) x, and therefore u — <r = (1 — cr) x, 1 — u = (1 — cr) (1 — x) 
and du = (l — cr)dx; then the limits (1, 0) of x correspond to the limits (1, a) of u, 
and the formula becomes 
V = r !-w//r--v f dt.t-y- 1 {(i + / 2 ).. .{t + -<r)“9 _1 I dx. x-v- 1 (f) [<r + (1 — a) x], 
1 (- <?) i (£ s + q)J e • o 
, q2 q2 
where a is retained in place of its value + • • • + + j • This is, in fact, a 
form (deduced from Boole’s result in the memoir of 1846) given by me, Cambridge 
and Dublin Mathematical Journal, vol. II. (1847), p. 219, [44]. 
If in particular <f)U = (1 — u) q+m , then (/> [a + (1 — cr) x) = (1 — cr) q+m (1 — x) q+m , and 
thence 
I* 1 x~ q ~ l {<f)tr + (1 -a) x) dx = (\- a) m f x- q -' (1 - x) q+m dx, 
_ r (— g )r(l+ g + m) _ 
T (1 + m) y } 
and then, restoring for cr its value, we have 
f - fK 1 ) (f - h) S? Ltrq ~'v p +t 
as the value of the integral