132 
A MEMOIR ON DIFFERENTIAL EQUATIONS. 
[655 
We have in fact 
and thence 
*•-*{- 7® (*-*•>+7(fj- (*+«•)}■ &c - ; 
p 2 + q 2 + r 2 = % {R + S + 2 (x 2 + y 2 + z 2 - x 2 — y 0 2 — z 2 )}, = x 2 + y 2 — z 2 , 
p<? + q<? + r 0 2 = {E + S - 2 (or 2 + y 2 + z 2 - x 2 - y 0 2 - z 0 2 )}, = x 0 2 + y 0 2 + z 0 2 , 
or the last-mentioned results are thus verified. 
Partial Differential Equation containing the Dependent Variable: Reduction to Standard 
Form. Art. Nos. 80, 81. 
80. The equation H = const, is the most general form of a partial differential 
equation not containing the dependent variable V; but if a partial differential equation 
does contain the independent variable, we can, by regarding this as one of the dependent 
variables, and in place of it introducing a new independent variable, exhibit the 
equation in the standard form H = const., H being here a homogeneous function of 
the order zero in the differential coefficients. Thus, if the independent variables are 
x, y, the dependent variable z, and its differential coefficients p, q, then the given 
partial differential equation may be H, = H (p, q, x, y, z), = const. But we may 
determine z as a function of x, y by an equation V = const., V being a desired 
function of x, y, z; and then writing p, q, r for the differential coefficients 
dV 
dV dV 
dx ’ dy ’ 
7) Q 
, , we have p = — ~, q = — -, and the proposed partial differential equation becomes 
CLZ 7* 7' 
H (- f > ~l> V> z ) = const. 
viz. this is an equation containing only the differential coefficients p, q, r of the 
dependent variable V, a function of x, y, z. And, moreover, H is homogeneous of 
the order zero in p, q, r\ consequently 
dH dH dH 
Pfy + iT q +r W = 0 ’ 
dV 
or, in the augmented Hamiltonian system, the last equality is = , so that an 
integral is V — const.; as already stated, this is the equation by which 5 is determined 
as a function of x, y. 
81. Thus, if the given partial differential equation be pq — z = H, we here consider 
the equation ^ — z = H. The Hamiltonian system is 
r 2 dx _r 2 dy _ — r 3 dz _dp dq _dr ( 
q p 2pq 0 0 1 \ 0 ) ’