774 
PROFESSOR CAT LEY ON PREPOTENTIALS. 
portion of the strip ; the integral is therefore to be extended only over the unshaded 
portion of the strip ; viz. the value is 
v- ffW-.ft) 
T(-q)T(is + q) 
dll dt . 
the double integral being taken over the unshaded portion of the strip ; or, what is the 
same thing, the integral in regard to u is to be taken from u — ( sa Y 
r+t 
from u—g ) to u— 1, and then the integral in regard to t is to be taken from t=d to 
t— oo, where, as before, 0 is the positive root of the equation < 7 = 1 , that is of 
rp P (P 
P+Q ^A*+r Q 
162. Write u=G-j-(l — g)x, and therefore u— <r=(l — g)x, 1— u—(l — <r)(l — x) and 
du=(l — a)dx\ then the limits (1, 0) of x correspond to the limits (1, g) of u, and the 
formula becomes 
Y= r(?;)rar+ g) r^- f'V+f’-t+vrK <>{H-(l-r> 
where a is retained in place of its value ~ — . .. + 4 ^+t- This i n a form 
J z ~rt tl z 4 * t 
(deduced from Boole’s result in the memoir of 1846) given by me, Cambridge and 
Dublin Mathematical Journal, vol. ii. (1847), p. 219. 
If in particular <pu=(l— u) q+m , then <p{cr+(l— <r)#}=(l — G) q+m (l— x) q+m , and thence 
f x~ q ~ l {<pff + (l— G)x}dx—{1— G) m ( x~ q ~\l—x) q+r 
Jo Jo 
r (-g)r(i+g +w) 
~ r(i+?w) 
and thence restoring; for g its value, we have 
n dx. 
fi °) m > 
• *^-(*+/*...*+v)-(i -j, 
-\-t 
h 2 +t t 
( a ? 2 z iy+m 
V-p-'-w) dx - dz 
\(a-x)*... + {c-z)* + e*\ is+q 
. This is in fact the thee 
its general form ; but the proof assumes that q is positive. 
over the ellipsoid A.„.-j-£_ = l. This is in fact the theorem of Annex IV . No. 110 in 
J h 
