68 
[725 
725. 
NEW FORMULAE FOR THE INTEGRATION OF 
[From the Messenger of Mathematics, vol. vm. (1879), pp. 60—62.] 
I have found in regard to the differential equation 
+ dy = 0 
f(a — x.h — x. c — x.d — x) f(a — y.b — y.c — y.d — y) ’ 
a system of formulae analogous to those given, p. 63, of my Treatise on Elliptic 
Functions, for the values of sn(w + 0, cn (u + v), dn (u + v). Writing for shortness 
a, b, c, d — a — x, b — x, c — x, d — x, 
a u K c,, d! = a-y, b-y, c-y, d-y, 
and (be, ad) to denote the determinant 
1, x + y, xy , 
1, b + c, be 
1, a + d, ad 
and (cd, ab), (bd, ac) to denote the like determinants; then the formulae are 
f(a — b.a — c) {VCadbjCj) + VOidJbc)} 
(be, ad) 
_ f(a — b .a — c)(x — y) 
VCadb^j) — -v/Oidjbc) ’ 
_f(a — b.a — c) |y'(abc 1 d 1 ) + VOifycd)} 
(a — c) VCbdbidj) - (b — d) //(aca^) ’ 
_ f(a — b .a — c) {VCacbidj) + y'OjCjbd)} 
(a — b) -v/Odcjdj) — (c — d) vXabajbj) ’