2 ON SOME INTEGRAL TRANSFORMATIONS. [l59 
and we have 
K (x — cl) = D (x' — d') ; 
where 
D = {o' - ct') (a' - 6') (6' - c') A, 
and 
. _ (a — d) (b — c) _ (b — d)(c — a) _ (c — d) (a — b) 
~ {a! — d') (b' — c') ~ (b'-d') (c'-a') “ (c - d') (a'- b') ‘ 
Suppose a + /3 + 7 + S= — 2; then 
(x — a) a (x — (¿c — c)y (x — d) s dx = J (x' — ci'Y (x' — b'Y {x' — c') y {od — d ) s dx, 
where 
J ={b — c)^ + v +1 (c — a)Y +a+1 (a — b) a+p+1 (b' — c')“ 4 " 54 * 1 (c' — a'Y +s+1 (a' - b') y+s+1 D s . 
We may in particular take for a', b\ c', d' the systems b, a, d, c : c, d, a, b and d, c, b, a 
respectively ; this gives, writing successively y, z, w instead of x', 
(x — a) a (x — b) p (x — c)y (x — d) s dx 
= M (y - af {y - by {y - c) s (y - d)y dy 
= N (z — a)y (z — b) s (z — c) a (z — dy dz 
= P (iv — a) s (w — b)y {w — c'y (w — d) a dw, 
where 
M — — (—)y+ 5 (a — c) a+Y+1 (a — dy +s+1 (b — cy +y+1 (b — df +s+1 , 
N = (—) Y+5 (a — b) a+li+1 (a — d) a+s+1 (b — c)^ +Y+1 (c — cZ)y +s+1 , 
P — (a — b) a+p+1 (a - c) a+Y+1 (b — dy +s+1 (c - d) y+s+1 ; 
the relations between the variables x, y, z, w being 
_ (c + d) ab — (a + b) cd — (cib — cd) y 
ab — cd — (a+ b — c — d) y 
(b + d) ac — (a + c) bd — (ac — bd) z 
ac — bd — (a + c — b — d)z 
_(b + c) ad — (a + d)bc— (ad — be) iv 
ad — bc — (a + d—b — c)w 
these are, in fact, the formulae in my note, “ On an Integral Transformation,” Camb. 
and Dubl. Math. Jour. t. ill. (1848), p. 286 [62], which was suggested to me by 
Gudermann’s transformation for elliptic functions, (Crelle, t. xxm. (1846), p. 330). 
Suppose now that the values of a', b', c', d' are 0, 1, oo, £, we have in this case 
a (6 - c) + c (a - b) y 
(b — c) + (a — b) y ’