41] 
TO THE EVALUATION OF DEFINITE INTEGRALS. 
269 
where 6 extends only from 6 = 0 to 9 = s, on account of the factor V {6 — s) -r- T (— s), 
which vanishes for greater values of 6: a rather better form is obtained by replacing 
this factor by 
( \e F (1 + s) 
K ' T(l + s — 6)' 
The above formulae have been deduced on the supposition of % being an integer; 
assuming that they hold generally, the equation (2) gives, by writing (i - \) for i, 
or integrating (i + |) times by means of the formula 
f o xi ~ l f x dx = (/ da ) + V g > «=°; 
this gives 
r ^dx rir(l-|) 1 .... 
J, ((x+X) (x + fi)y Fi (Vx + vW ai-1 ’ 
whence also 
r _rjr(i+i) 1 
J o 0 + ^y +1 (x + ^y r (* +1) {sjx + V/*) 2 v*. w ’ 
and from these, by simple transformations, 
" a (a — opy-i (x — ¡3)^ dx T r (i + §) (a — /3y 
r /3 {(a — x) + m (cc — /3)Y r(^+l) {*Jm + l) 2i " 
; a (g - *)*-» < x - py-t dx r I r (% - 1) (a- /3)^ ^ 
f js {(a — x) + m {ac — /3)} i Ti (Vra + l) 2i_1 ^ 
These last two formulae are connected also by the following general property: 
“ Tf (n h A — f a ( g ~ x ) a 1 ( x ~ fi) b 1 dx 
{ ’ ’ } J? {(a — x) + m(x — /3)Y ’ 
then (*> h > V Vi (a “ ^ ( a + b ~ i > £)” ( 8 )> 
which I have proved by means of a multiple 2 integral. From (6) we may obtain for y< 1, 
f 1 (1-<*)*-* d® r(*)r (» + *) ^ 
J_ 1 (l-2 7 ® + 7 2 ) < r(*+l) 1 
1 This is immediately transformed into 
f x rjr(i-i) 1 
J 0 (ax 2 + bx + c) 1 Tf {ZH-2 v /(ac)} i_ i ’ 
which is a particular case of a formula which will be demonstrated in a subsequent paper. [I am not sure to 
what this refers.] 
2 [The triple integral j Jj ui 1 ® a 1 1 e (%+ m y)u-x Vdxdydu.]