A MEMOIR ON DIFFERENTIAL EQUATIONS. 
119 
[655 
are themselves 
as functions of 
i, b, c, are the 
;heir values as 
the system as 
), r 0 , t 0 be cor- 
..., V 0 be the 
} variables ; we 
le, 
:e. 
: o! 
e 0 , we express 
constants, then 
655] 
The Partial Differential Equation H — const. Art. Nos. 59 to 70. 
59. In what just precedes we have, in fact, brought the theory of the Hamiltonian 
system into connexion with a partial differential equation, viz. we have determined 
the variables p, q, r as functions of x, y, z such that p dx + qdy + rdz is an exact 
differential = dV; but we now consider the subject in a more regular manner. 
The partial differential equation is H = const, viz. here H denotes, in the first 
instance, a given function of p, q, r, x, y, z, where p, q, r are the differential 
coefficients of a function V of x, y, z, or, what is the same thing, there exists a 
function V of x, y, z such that p dx + qdy + r dz = dV; and then, this function H 
being constant, we use the same letter H to denote the constant value of the 
function. The equation H = const, is the most general form of a partial differential 
equation of the first order which contains the independent variable only through its 
differential coefficients p, q, r, and it is for convenience put in a form containing 
the arbitrary constant H, which constant might without loss of generality be put = 0 
or = any other determinate value. 
60. We seek to determine p, q, r as functions of x, y, z, satisfying the given 
equation H = const., and such that we have p dx+ qdy + r dz an exact differential 
= dV; this would be done if we can find two other equations K = const, and 
L = const., such that the values of p, q, r obtained from the three equations give p, q, r 
functions having the property in question. Attending to only two of the equations, 
say H= const, and K = const., we have here p, q, r functions of x, y, z, such that 
pdx + qdy + rdz is an exact differential, and two of the equations which serve to 
determine p, q, r as functions of x, y, z are H = const., K = const. We have to 
prove the following fundamental theorem, viz. that (H, K) = 0. 
61. In fact, from the equations H = const., K = const., treating x, y, z as inde 
pendent variables, we have 
dH dH dp dH dq dH dr 
— 1 -\ -A -j — = 0 
dx dp dx dq dx dr dx ’ 
dK dK dp dK dq dK dr _ ^ 
dx dp dx dq dx dr dx ’ 
and 
add, 
if from these equations in order to eliminate 
we find 
dp 
dx 
we multiply by 
dK 
dp : 
d(K, H) d (.K, H) dq d(K, E) dr 
d {p, x) ' d (p, q) dx d (p, r) dx 
dH 
-T— , and 
dp 
and, in precisely the same way, 
d(K, II) d{K, H) dp d(K, H) dr = 0 
d (q, y) d (q, p) dy ‘ d (q, r) dy~ ’ 
d{K, H) d(K, H) dp d(K, H) dq | =0 
d (r, z) d (r, p) dz d (r, q) dz 
on H= const.