■i.. mnmmmm 
14 ON CERTAIN DEFINITE INTEGRALS. [103 
o . r\ / ■ \ 9 i x + ill) fix + iy) 
ry(x + ly) 4> (x + ly) 
4. + , + , © (x + iy) = 7 * ; 
v(* + *y) 
where <£, f, F are in fact the symbols of the inverse elliptic functions (Abel’s notation) 
corresponding very nearly to sin am, cos am, A am. It is remarkable that the last 
value of © cannot be thus expressed, but only by means of the more complicated 
transcendant <yx, corresponding to the H (x) of M. Jacobi. The four cases correspond 
obviously to 
1. '\fr(x + nv, y -+ sv) = (—) r+s -v/t (x, y), 
2. yjr (x + rw, y + sv) = {-Y yfr(x, y), 
3. yjr (x + rw, y + sv) = (—) s \Jr (x, y), 
4. y\r {x + rw, y + sv) — (x, y). 
The above formulae may be all of them modified, as in the case of single integrals, 
by means of the obvious equation 
^ (9 X > gy) dxdy 
0 (x + iy) dxdy = wv 
JJ (X + iyY „„ 
The most important particular case is 
r 00 g 
J-acj-x, (x + iy) 
for in almost all the others, for example in 
the second integration cannot be effected. 
f d y- 1 
1 
\dxj 
(j)(x + iy)_ 
dxdy, 
Suppose next (%, y) is one of the functions y (x + iy), g (x + iy), G(x + iy), 
(S (x + iy), so that 
yjr (x + rw, y + sv) = (±) r (±) s (x, y), 
where 
Ur,, = (-) rs F XiTW ~ svi) q-^ q-№, 
(see memoir quoted). Then, retaining the same value as before of T" (x, y), we have 
still the formula (B), in which 
(±) r (±) s Ur,s 
© (x + iy) = 2 
x + iy + rw + SVl 
But this summation has not yet been effected; the difficulty consists in the .variable 
factor €^ x i rw ~ svi) in the numerator, nothing being known I believe of the decomposition 
of functions into series of this form.