656] 
ON THE THEORY OF PARTIAL DIFFERENTIAL EQUATIONS. 
13 7 
(2H 
or since (H, a) = 0, (b, H) = 0, the left-hand side is simply 
becomes 
d(c 0 , d 0 , e 0 ) if dr dp\d(a, b, H) dH 
dq 
(a, b), and the equation 
d(p 0 , q 0 , r 0 ) (\dx dz) d(r, p, q) 
- d ^(a, &)} + &c. = 0. 
dv dj) 
This ought to give = 0, and it will do so if only 
fa(c°, d 0 , e 0 ) 6)1 = 0; 
(9 (p 0 , q 0 , n) J 
this is then the equation which has to be proved. By the Poisson-Jacobi theorem, (a, b) 
is a function of a, b, c, d, e: if we write 
/ 3(oo, 6 0 ) 3(a 0 , b 0 ) d(a 0 , b 0 ) 
{Ct °’ 0) ~d(p 0 , xyd{q 0 , y 0 ) + d(r 0 , z o y 
then (a 0 , b 0 ) is the same function of a 0 , b 0) c 0 , d 0 , e 0 ; but these are = a, b, c, d, e 
respectively, and we thence have (a, b) = (a 0 , b 0 ), and the theorem to be proved is 
0 (C 0 , da, 6a) 
3 {po, qo, n) 
(«0, 9o)r = o. 
But substituting for (a 0 , b 0 ) its value, the function on the left-hand is (it is easy to see) 
the sum of the three functional determinants 
9 (fla, b 0 , Cq, d 0 , 6a) d (tto> bo, Co, da, 6a) ^ 0 (tto> bo, Cq, d 0 , 6q) 
3(po, q 0 , r o, Po, Xo) d(p 0 , q 0 , r 0 , q 0 , y 0 ) d(p 0 , q 0 , r 0 , r„ z 0 )’ 
each of which vanishes as containing the same letter twice in the denominator, that is, 
as having two identical columns; and the theorem in question is thus proved. And in 
the same way ^ ~ ^ are each = 0 : or we have p dx + qdy + rdz an exact 
differential. 
The proof would fail if the factors multiplying ^ ™, &c., or if any one of these 
factors, were = 0; I have not particularly examined this, but the meaning would be, 
that here the equations in question a = a 0 , &c., H = const., are such as not to give 
rise to expressions for p, q, r as functions of x, y, z, x 0 , y 0 , z 0 , H, as assumed in 
the theorem ; whenever such expressions are obtainable, then we have p dx + q dy + r dz 
an exact differential. 
The proof in the case of a greater number of variables, say in the next case 
where the variables are x, y, z, w, p, q, r, s, would present more difficulty—but I have 
not proceeded further in the question. 
It is worth while to put the two processes into connexion with each other: taking 
in each case the variables to be x, y, z, p, q, r, and the partial differential equation 
to be H = const.; 
C. X. 
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