[103 
103] 
ON CERTAIN DEFINITE INTEGRALS. 
13 
becomes infinite for 
incipal value, that is, 
>r which the quantity 
scent. 
x~ fl , we have finally 
;he latter to that of 
- x, then ‘ÿ / (x+ 7r) 
Id. Combining these 
cot x 
dx ; 
cosec x 
dx ; 
d y 1 
dx) 
In particular 
/' 
sin xdx 
= 7T, 
sin # (—)**' 1 f n 
J dx ;— I sin x 
gx 
( — 
cot X 
af- 
dx 
S’ 
/>4(; 
\dxj 
I w-i ~| r # 
cosec a; die = g** -1 J sin 
I sin gx cot xdx= tt, g even, 
J 0 
~fd 
\dx/ 
cosec x 
~( d - 
P.— 
\dx/ 
cosec x 
~(d 
P.—Ì 
\dxj 
cosec x 
dx, 
dx, g even, 
dx, g odd, 
sin gx cosec xdx = 7r, <7 odd, 
^taniedie . 0 
= 0, &c., 
the number of which might be indefinitely extended. 
The same principle applies to multiple integrals of any order: thus for double 
integrals, if y}r(x + ra, y + rb) = TJ r s yjr (x, y), then 
[ I ^y)¥y) dxdy = f f yfr (x, y) XU rtS y ¥(x + ra, y + sb). ... (B) 
J — ao J — 00 J 0 J 0 
In particular, writing w, v for a, b, and assuming yjr(x + rw, y + sv) = (±) r (±) s yjr (x, y)\ 
also 'F (x, y) = (x + iy) _ ' i , where as usual i = V — 1, 
where 
' f * ylr (x, y) dxdy (-y- 1 f w f v . , s 
(±)'(±)*1 
dxdy, ...(B') 
® (a; 4- fi/) = X 
(# + iy +rw + svi) ’ 
X extending to all positive or negative integer values of r and s. Employing the 
notation of a paper in the Cambridge Mathematical Journal, “On the Inverse Elliptic 
Functions,” t. iv. [1845], pp. 257—277, [24], we have for the different combinations 
of the ambiguous sign, 
„ / . (S {x + iy) 1 
, — , © (x + iy) = —7—-- -- = , 7——i-t , 
v <y(x-[-iy) (f)(x + iy) 
1. 
2. 
„ . . . G (x 4- %y) F (x + xy) 
+ , <d(x + iy)= 7 ;—r-^ = ~T7 \—• \ 
v 7 (x + iy 4> (x + iy) 
cosec x 
dx.