ON CERTAIN DEFINITE INTEGRALS. 
15 
3] 
the limits of the first side of the equation, and the condition to be satisfied by a, b, 
&c., also the equation for the determination of £, being as above. 
The integral 
V' = [[... (n times) dxdy... 
between the same limits, and with the same condition to be satisfied by the constants, 
has been obtained [see p. 11] in the paper already quoted. Writing £ instead of rf, and 
f 
?+■#> 
, we have 
V' = 
hh,... 7r in r 
dcf) 
where 
l' 1 Qn) Jo (£ + $) V{(£ + A 2 + 0) (£ + hf + 4>)...} ’ 
= 1. 
a 2 6 2 
f + A 2 + F + v 
T _ d 2 d 2 
Let V ~da? + d¥ + 
Then by the assistance of a formula, 
V 1 (o. + y...)< = 2i + 2) • • • ( 2i + 2? - 2) (2* + 2 - n) ... (« + 2 4 - n). (ga + y -yr, 
given in the same paper [see p. 6], in which it is obvious that a, b... may be changed 
into a — cc, b — y, &c....; also putting ¿ = |n; we have 
„ in 
hh. 
d<b. V q 
(£+tf>)\/{(M-A 3 +4>)...}' 
//•■• (n times) t.rau+dj. 
Now in general, if ^ be any function of £, 
But from the equation 
we obtain 
whence 
v — I 
(l+A 2 ) ’ 
2a 
FT 
- 2 
i 2 | d£ _ 
(£ +A 2 ) 2 ] da 
0, 
v ÆfV = 
da/ 
(£ + a 2 ) 2 
Also 
2 _ 4_- a t .. .^I + 2 |2 
£ + A 2 (f + A 2 ) 2 da 
whence taking the sum 2, and observing that 
/-.2 
N. 
a 2 
a 2 1 d 2 £ 
(£ + A 2 ) 2 ] da 2 
= 0; 
qv a _ 8 
(Ç + hJda 
22 
(g + A 2 ) 3 
a 2 
(ITT 2 ) 2 
= -22 
(£+A 2 ) ! 
. 2 
d|y 
da/ ’ 
l N (Ì£\ = 0; 
£ + A 2 ( (|+A 2 ) 2 ] Vdd