A MEMOIR ON DIFFERENTIAL EQUATIONS. 
121 
[655 
655] A MEMOIR ON DIFFERENTIAL EQUATIONS. 121 
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and, if it vanish, it is impossible, by means of the equations H = 0, K = 0, L = 0, 
to determine p, q, r as functions of x, y, z. Hence, if the last-mentioned equations 
are such that by means of them it is possible to effect the determination, and if, 
moreover, (H, K) = 0, {H, L) = 0, (K, L) = 0, then pdx + qdy + rdz will be an exact 
is the theorem 
differential. 
if (K, H) = 0, 
64. Considering H as given, we have, by what precedes, K, L solutions of the 
linear partial differential equation (.H, 6) = 0 ; and since also K, L must be such 
that (K, L) = 0, they are conjugate solutions ; or in conformity with what precedes, 
using the small letters a, b instead of K, L, we have the following theorem for the 
integration of the partial differential equation H = const., where as before H is a 
given function of x, y, z, p, q, r. 
i is so unless 
Find a and b, such that H, a, b are a system of conjugate solutions of the linear 
partial differential equation (H, 6) = 0 : then from the equations H = const., a = const., 
actions of p, q, 
b = const., determining p, q, r as functions of a, b, H. and in the result treating* 
could not from 
eliminating one 
ace obtain the 
Deing such that 
ictions of x, y, 
these quantities as constants, we have pdx + qdy + rdz an exact differential =dV, 
and thence 
V=X+ j"(p dx + qdy + r dz), 
an expression for V containing the three arbitrary constants a, b, and therefore a 
complete solution of the given partial differential equation H = const. 
if p, q, r are 
0, L = 0, then, 
differential, we 
3 that if these 
The theorem applies to the case where n has any value whatever, viz. if there 
are n variables x, y, z,, then we have to find the n — 1 integrals a, b, c,... y 
constituting with H a system of conjugate integrals ; and the theorem holds good. 
In particular, if n = 2, or the independent variables are x and y, then we find any 
solution a of the partial differential equation (H, 6) = 0 ; the values p, q derived from 
the equations H = const., a = const., give V=\ + j'(pdx + q dy), a complete solution. 
unless we have 
65. But there is a different solution depending on the consideration of corre 
sponding values; viz. if the independent variables be as before x, y, z, p, q, r, and if 
x 0 , y 0 , z 0 , p 0 , q 0 , r 0 are corresponding values of x, y, z, p, q, r, then, taking a, b, c, d, e 
to be integrals of (H, 6) = 0 : so that H is here a given function of a, b, c, d, e, 
since the number of independent variables is =5: and representing by a 0 , b 0 , c 0 , d 0 , e ( > 
the like functions of x 0 , y 0 , z 0 , p 0 , q 0 , r 0 , we form the equations 
H = const., a = a 0 , b = b 0 , c = c 0 , d — d 0 , e = e 0 . 
ie functions of 
We have the theorem that, expressing by means of these equations p, q, r, as 
functions of x, y, z, x 0 , y 0 , z 0 , H, and regarding therein x 0 , y 0 , z 0 , H as constants, 
we have p dx + qdy + r dz an exact differential, and therefore 
V = X + (pdx + q dy + r dz), 
C. X. 
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