PROFESSOR CAYLEY ON PREPOTENTIALS. 
773 
general differentiation, we have the result in question. He has in the formula - instead 
of my t ; and he proceeds, “ Here <r increases continually with s. As s varies from 0 to oo , 
<t also varies from 0 to oo . To any positive limits of <r will correspond positive limits 
of s ; and these, as will hereafter appear [refers to his note B], will in certain cases replace 
the limits 0 and oo in the expression for V.” 
160. It seems better to deal with the result in the following manner, as in part shown 
p. 803 of Boole’s memoir. Writing the integral in the form 
V = (^1 + J dudt dv . v q cos|(m— 
effect the integration in regard to v ; viz. according as u is greater or less than <r, then 
f" 7 „ \ . i i r (<7 + 1) sin {q-\-\)n n 
l dv . p g cos{(M— <7)v-\-\qr\= — . q+1 — — , or 0, 
Jo \ U G" ) 
and consequently, writing for a its value, 
7 r 
= r(-g)( W -<r)^ 1 
, or 0 ; 
P + t'" h* + t t' ® U ’ 
or 0 as above k 
Y== r(-;)r»+ g) j’J, du dt 
161. To further explain this, consider t as an ^-coordinate and was a ^-coordinate; 
«=. “ .. .4-— L 
J P+x 
for positive values of x, this is a mere hyperbolic branch, as shown in the figure, viz. 
#=0, y=co ; and as x continually increases to oo , y continually decreases to zero. 
The limits are originally taken to be from u=0 to u — 1 and £=0 to £=oo , viz. over 
the infinite strip bounded by the lines £0, 01, 11 ; but within these limits the function 
under the integral sign is to be replaced by zero whenever the values u, t are such that 
« 2 c 2 e 2 
u is less than viz. when the values belong to a point in the shaded 
