7] ON A CLASS OF DIFFERENTIAL EQUATIONS, &C. 37 
Now from the equation (1), we have the system 
fx +gy + hz = 0 (6), 
fdx + gdy -f hdx =0, 
fd n ~-x + gd n ~ 2 y + hd n ~-z = 0, 
or writing X= y , z , (7), 
dy , dz , ... 
d n ~ 2 y, d n ~ 2 z, ... 
with analogous expressions for Y, Z ; then from the equations (6),/, g, h are 
proportional to X, Y, Z : or, eliminating by (2), 
H(X, Y, Z ) = 0 (8). 
Conversely the integral of the equation (8) of the order (n - 2) is given either bv 
the system of equations (1), (2), in which it is evident that the number of arbitrarv 
constants is reduced to (n^- 2); or, by the equation (5), which contains in appearance 
n (ft — 2) arbitrary constants, but which we have seen is equivalent in reality to the 
system (1), (2). 
Thus, with three variables, the integral of 
11 (ydz — zdy, zdx — xdz, xdy — ydx) = 0 (9) 
may be expressed in the form 
H {yzi “ Vi z > zx i ~ z i®, ~ any) = 0 (Kl), 
and also in the form fx + gy + hz=0 (ll) ? 
where H (f g, A) = 0 (12). 
The above principles afford an elegant mode of integrating the differential equation 
for the lines of curvature of an ellipsoid. The equation in question is 
(6- — c‘ 2 ) xdydz + (c 2 — a 2 ) yd zdx + (a 2 — b 2 ) zdxdy = 0 (13), 
where x, y, z are connected by 
writing 
X 2 
a 2 
(15), 
these become 
(l) 2 — c 2 ) udvdw + (c 2 — a 2 ) vdwdn + (a 2 — b 2 ) wdudv = 0 
u+ v + w = 1 
(16), 
(17).