46 
a smith’s prize paper, 1877. 
[645 
where A, /x, v are arbitrary ; viz. multiplying by £, rj, £, and adding we have the 
equation in question. The three equations determine f, rj, £ as linear functions of 
x, y, z; and we have thence x x , y 1} z x as linear functions of x, y, z\ viz. this is a 
solution containing three arbitrary constants A, p, v. 
12. The partial differential equation might equally well have been proposed in the 
form, given function of x, y, z, p, q, r= 0, viz. the equation then is </> (x, y, z, p, q, r) = 0, 
the general partial differential equation involving the three independent variables x, y, z, 
and the derived functions p, q, r of the dependent variable u, but not involving the 
dependent variable u. The question is therefore in effect as follows: to find p, q, r 
functions of x, y, z connected by the foregoing equation, and, moreover, such that 
pdx + qdy + rdz is an exact differential ; for then writing U =J(pdx + qdy + rdz), we have 
the solution of the given partial differential equation. 
Whatever be the method adopted, it comes out that the solution depends on the 
integration of the system of ordinary differential equations 
dp _ dq _ dr _dx _ dy _ dz 
d(f> d(f> d<f) d(f) d(j) d<f> ’ 
dx dy dz dp dq dr 
and the answer consists first in showing this, and secondly, in showing how from an 
integral or integrals of the system we pass to the solution of the partial differential 
equation. 
Considering the partial differential equation in the form actually proposed, we may 
instead of <f> write H, where H will stand for that given function of x, y, z, p, q, r which 
is the value of the arbitrary constant H ; making this change, and putting the fore 
going equal quantities equal to the differential dt of a new variable, the system of 
ordinary differential equations is 
dp 
dH 
dq 
dH 
dr 
dH 
dt 
dx ’ 
dt 
dy 5 
dt 
dz 
dx 
dH 
dy^ 
dH 
dz 
dH 
dt 
dp ’ 
di 
dq ’ 
dt 
dr 
where H is a given function of x, y, z, p, q, r. This is, in fact, the Hamiltonian system 
of equations; and it was in view to the connexion that the partial differential equation 
was proposed in its actual form.