[656
656] ON THE THEORY OF PARTIAL DIFFERENTIAL EQUATIONS. 135
lUATIONS.
«.]
ction of certain
articular case.
3s is, in fact, a
lies): and when
the function in
atial coefficients
(as a letter not
; it as a given
instance, if H
i, pqr — xyz = H
Is upon that of
we determine p, q, r as functions of x, y, z, the resulting value of pdx+qdy+rdz is
an exact differential, and we have
V = X + I (pdx + qdy + r dz),
a solution containing three arbitrary constants, X, a, b, and therefore a complete solution
of the proposed partial differential equation H = const.
But (as is known) there is a different process of integration, for which the con
jugate solutions are not required, and which has reference to a system of initial values
x 0 , 2/ 0 , z 0 , p 0 , q 0 , r 0 : viz. if the independent solutions of (H, ©) = 0, are a, b, c, d, e, and
if a 0 , b 0 , c 0 , d 0 , e 0 denote respectively the same functions of the initial variables that
a, b, c, d, e are of x, y, z, p, q, r, then if from the equations
a = a 0 , b = b 0 , c = c 0 , d = d Q , e = e 0 , H = const.
we express p, q, r as functions of x, y, z and of x 0 , y 0 , z 0 , H, these last being regarded
as constants, we have pdx + qdy+rdz an exact differential, and
F = X + j (pdx + qdy +r dz),
a solution containing the constants X, x 0 , y 0 , z 0 (that is, one supernumerary constant), and
as such a complete solution.
It is interesting to prove directly that pdx + qdy + rdz is an exact differential.
I consider first the more simple case where the variables are p, q, x, y. Here p, q
are to be found from the equations
a = a 0 , b — b 0 , c = c 0 , H = const,
and it is to be shown that pdx + qdy is an exact differential.
Considering p, q, p 0 , q 0 as functions of the independent variables x, y, then dif
ferentiating in regard to x, and eliminating we have
da
l
da
dq
da
da 0
da 0
dx
T
dq
dx ’
dp ’
dp 0 ’
dq 0
db
+
db
dq
db
dbo
db 0
dx
dq
dx’
dp ’
dpo ’
dq 0
dc
+
dc
dq
dc
dco
dc 0
dx
dq
dx’
dp’
dp 0 ’
dq 0
dE
dx
+
dE
dq
dq
dx ’
dE
dp ’
0 ,
0
ion (H, ©) = 0,
gate solutions)
