118 
A MEMOIR ON DIFFERENTIAL EQUATIONS. 
[655 
4m 
and again treating a, b, H as variable, we have 
dV—pdx—qdy— rdz = d\ +(t — t) dH + Ada + Bdb, 
where A, B are functions of the integrals a, b, c, d, e, that is, they are themselves 
integrals, which may be taken for the integrals d, e, or we have 
dV — p dx— qdy — r dz — dX + (£ — t) dH + dda + edb; 
we have therefore 
dV_ dV 
da ~ d} db 
equations which, on substituting therein for a, b, H their values as functions of 
x, y, z, p, q, r, determine the integrals d, e, which with a, b, H or a, b, c, are the 
remaining integrals of the Hamiltonian system; and further 
which, when in like manner, we substitute therein for a, b, H, their values as 
functions of x, y, z, p, q, r, determines r, the remaining integral of the system as 
increased by the equality = dt. 
58. Reverting to the general theorem No. 52, let x 0 , y 0 , z 0 , p 0 , q 0 , r 0 , t 0 be cor 
responding values of the variables x, y, z, p, q, r, t; and let a 0 , &c., ..., V 0 be the 
same functions of x 0 , y 0 , z 0 , p 0 , q 0 , r 0 , 4 that a, &c.,..., V are of the variables; we 
have a = a 0 ,..., e = e 0 , and corresponding to the equation 
dV —pdx — qdy —rdz = d\ + (t — r) dH + Ada + ... + Ede, 
the like equation 
dVo-p 0 dx 0 — q 0 dy 0 - r 0 dz 0 = d\ + (t 0 — r) dH + Ada + ... + Ede. 
Hence, subtracting 
dV - dV 0 = (t — 4) dH +p dx + qdy + r dz — p 0 dx 0 — q 0 dy 0 — r 0 dz 0 , 
or, considering only H as an absolute constant, 
dV-dV 0 = 
pdx + qdy + rdz-p 0 dx 0 - q 0 dy Q - r 0 dz 0 ; 
viz. if from the equations H = const., a = a 0 , b = b 0 , c = c 0 , d = d 0 , e = e 0 , we express 
P, q, r > Po, q», r 0 as functions of x, y, z, x 0 , y 0 , z 0 , H, then 
p dx + qdy + r dz - p 0 dx 0 — q 0 dy 0 — r 0 dz 0 
will be an exact differential. And in particular regarding x 0 , y 0 , z 0 as constants, then 
p dx + qdy + rdz is an exact differential, viz. there exists a function 
We have thus again arrived at a solution of the partial differential equation H= const. 
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