62] 
383 
62. 
ON AN INTEGRAL TRANSFORMATION. 
[From the Cavibridge and Dublin Mathematical Journal, vol. ill. (1848), pp. 286—287.] 
The following transformation, given for elliptic functions by Gudermann (Crelle, 
t. xxiii. [1842], p. 330) is useful for some other integrals. 
dbc — dba — dca + abc — (be — ad) z 
(be — ad) + (d—b — c +a) z ’ 
K = (be — ad) + (d — b — c + a) z, 
we have, supposing a <b < c < d, so that (b — a), (c — a), {d — b), (d — c) are positive, 
K (y — a) =(b — a) (c — a) (d — z), 
K (y - b) = (b - a) (d - b) (c - z), 
K (y - c) = {c - a) {d - c) (b - z), 
K (y — d) = (d — b) (d— c) (a — z), 
K 2 dy = — (6 — a) (c — a) (d — b) (d — c) dz. 
In particular, if a + ¡3 + <y + 8 = — 2, 
(y — a) a (y — by (y — c)y (y — d) 8 dy = — M(z — a) s (z — by (z — c) p (z — d) a dz, 
where M=(b — a) a+fi+1 (c — a) a+ y +1 (d — by +s+1 (d — c)* +5+1 . 
Thus, if a = /3 = 7 = S = — L 
{- (V ~a)(y~ 0 (V -c){y- djy ~ {-(z-a) (z -b)(z- c) (z - d)}*' 
In any case when y = a, y = b, the corresponding values of £ are z = d, z — c\ the 
last formula becomes by this means 
f b dy _ C d dy 
L {- (y ~ a) (y ~ 8) (y -c)(y- d)^ ~J c {-(z-a)(z-b)(z- c) (z - d)]* ‘ 
If 
then, putting