725] 
NEW FORMULA FOR THE INTEGRATION OF + 
vM- J Y 
69 
v 7 
b-z 
d — z. 
sj(~T^) K a ~ c)V(bdb 1 d 1 ) + (b -d)J(acaA)} 
(6c, aoZ) 
\/ (a - ¿) ^( abCldl ) ~ ^( a ibicd)} 
/^(adbiCj) — Viaidjbc) 
¿=3) (cd - ,l6) 
(a — c) v^bdbjd!) — (6 — cZ) ^(acaA) ’ 
a 
a ■ 
c — z 
d— z 
^) {(a — d) VibcbjCj) + (6 — c) A/Cadaidi)} 
(a — 6) Vicd^dj) — (c — d) VCabajbj) 
\/(a - d j :ia ~ ,; l V( c <fcid,) + (c - (/) v'(aba,bi)) 
V 
a — c 
a 
(be, ad) 
2) { / '/(acb 1 d 1 ) - V(a^bd)} 
VCadbjCj) — ^/(aidjbc) 
dj {(« — d) VlbcbjCj) — (6 — c) VCadajdj)] 
(a — c) /^(bdbidj) — (b—d) V(acajCi) 
d )(bd, ac) 
(a — 6) Vlcdcjdj) - (c — d)V(abaxbx) ’ 
The twelve equations are equivalent to each other, each giving ^ as one and the 
same function of x, y\ and regarding £ as a constant of integration, any one of the 
equations is a form of the integral of the proposed differential equation. 
Writing in the formulae x — a, b, c, d successively, the formulae become 
x = a, 
a — z 
d — z 
a x 
dx’ 
b — z _b 1 
d — z d x ’ 
X = 6, X = c, 
c — ft b] 6 — a c x 
d — b c 1 > d — c b x ’ 
c — 6 a x 6 — a.b — c dj 
d —a c x ’ d — a.d — c bj ’ 
x =d, 
b .a — c dj 
d — 6. d — c a. 
c — a. c 
b d, 
c a. 
eZ — £ d x ’ d — a. d — b c 1 ’ 
d — a b x ’ 
a — b Cx 
d — c aj 
a — c bj 
d — b a, ’ 
viz. in the first case we have z — y, and in each of the other cases 2 equal to a 
linear function of y. 
yy + 8 y 
Cambridge, July 3, 1878.