( 460 ) 
where Pand Q represent polynomials in y, he has proved that if 
the integrals possess m branches, there always exists a substitution 
me Oe: + Lit git = eee = Ly 
My ae 
by which the equation (1) may be reduced to an equation of 
Riccati 
u 
(2) 
. 
du 
Gat ee KE ne ec 
da 
the coefficients L, WM, G, H, K being functions of x. 
Our object in this paper is to prove this proposition in another 
way, starting from the form of the integral 
oe an YP + An gee -. thy +4, 
fg Sn ee ee Seat AY oe eg 
where C represents an arbitrary constant and 2 and u functions of z. 
The treatment of the two cases n=? and „=3 will be sufficient 
to show that the proposition holds good generally. 
(4) 
2. If n=2, it is evident from the integral 
Ay? + Ay + Aa 
C= a gie os nst on + i 1 
y? > BY > Bo 
that the differential equation must be of the form 
dy ay’ + ay’ Hay + ay +4, 
de by? + 2b,y + b, 
(6) 
the coefficients « and 5 representing functions of z. 
Differentiating the equation (5), we find between a, 0,4, u, the 
following relations 6 being an indefinite factor, 
Ce =e 
Oa, = wa,’ + a,’ — Au, 
Ga, = ud, Hud +4,’ — Au — At 
Oa, = ud, Hud, — Jol — Allo 
Oa == Uik =d 
0b, = 4, — ud, 
Ob == A = ba, 
Gb, == 1,4, — ots - 
From the three latter equations (7) may be induced 
