358 
[941 
941. 
NOTE ON THE PARTIAL DIFFERENTIAL EQUATION 
Rr + Ss + Tt + U (s 2 — rt) — V— 0. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xxvi. (1893), 
pp. 1—5.] 
It is well known that this equation, R, S, T, U, V being any functions whatever 
of (x, y, z, p, q), in the case where u admits of an integral of the form u=f(v) 
(u, v functions of x, y, z, p, q, and f an arbitrary functional symbol) can be integrated 
as follows; viz. taking m 1} ra 2 as the roots of the quadratic equation 
m 2 — Sm + RT — UV = 0, 
(that is, writing m 1 + m 2 = S and m 1 m„ = RT—UV), then, m 1 denoting either root at 
pleasure, and m 2 the other root of the quadratic equation, if the system of ordinary 
differential equations 
m x dx — Rdy + U dq = 0, 
— T dx + m 2 dy + U dp = 0, 
— V dx + m 2 dq + R dp = 0, 
— Vdy + T dq + m Y dp = 0, 
— p dx — q dy + dz = 0, 
(equivalent to three independent equations) admits of two integrals u = const, and 
v = const., the solution of the given partial differential equation is u = f{v). 
In fact, to prove this, we have 
du= \( m x dx — Rdy + U dq) 
+ p (— T dx + m 2 dy + U dp) 
+ v {— V dx + m 2 dq + R dp) 
+ p (— V dy + T dq + m x dp) 
+ o- (— pdx — q dy + dz),