PROFESSOR CATLET ON PREPOTENTIALS. 
711 
59. I consider also the case q= — l ; the integral is here 
j* a?-*(l - { 1 ■ - (1 )dx 
=e(X-i-l)+±e{ 1 x-$\l-(l -x)*- l \dx; 
Jx 
and the first term of this being =\/ e 2 +R 2 — e, this is consequently 
=\/^ 2 + 62 +i^r x-%\l — (1— xf s ~'\dx— c(l+| f x~l { 1 — (1 —xf s ~ 1 1 dx). 
Jo Jo 
As regards the second term of this we have 
-2x-^l-(l-x)^- 1 }+2(is-l)jx^(l-x)^- 2 dx=jx-i{l-(l-x)^- 1 ldx; 
or taking each term between the limits 1, 0, 
+ 
viz. this integral has the value 
o , 
T R ys-'dr 
Jo {r* + e 
a , is consequently 
=VR 2 +e 2 +^£ x~?{l — (l-xf~'\dx-e r(^z|y’ 
which is of the form 
1 
say the approximate value is 
R (l-f terms in . . A—e 
R — e 
R 2 ’ R 4 ’ 
nis-l) ’ 
where the first term R is the term dr, given by the expansion in ascending powers 
of e 2 ; the second term is the term in 0 ~ 2? . And observe that term is the value of 
x~i(l—xf s ~ l dx, 
Jo 
calculated by means of the ordinary formula for a Eulerian integral (which formula, on 
account of the negative exponent — §, is not really applicable, the value of the integral 
being =co ) on the assumption that the T of a negative q is interpreted in accordance 
with the equation T{q-\-^)=qTq ; viz. the value thus calculated is 
r(-i)r(i s ) _ 
2 r(is-i) ’ r(i«— i) 
on the assumption r^= — -|E( — ; and this agrees with the foregoing value. 
