A MEMOIR ON DIFFERENTIAL EQUATIONS. 
108 
[655 
and if we herein attend to the terms which contain the second differential coefficients 
of H, these are symmetrical functions of a, b. For instance, 
dH 
dx* ’ 
™ . . . da db 
coefficient is -7- -7- , 
dp dp 
dH 
da db 
da db 
dx dy 
dp dq 
dq dp ’ 
d?H 
da db 
da db 
dx dp 
dp dx 
dx dp ’ 
d?H 
da db 
da db 
dxdq 
dp dy 
dy dp ' 
Hence, forming the like terms of the second terms (b, (a, H)) and subtracting, the 
terms in question all vanish: and we thus see that (a, (b, H)) — (b, (a, H)) is a linear 
function of the differential coefficients 
dH dH dH dH dH dH 
dx ’ dy ’ dz dp ’ dq ’ dr 
dH 
36. Attending to any one of these, suppose , the coefficient of this 
in {a, (b, H)) is 
in (6, (a, H)) „ 
wherefore, in the difference of these, it is 
Hence, for the several terms 
dH dH dH dH dH dH 
dx dy dz dp ’ dq ’ dr 
the coefficients are 
( d d d 
\dp ’ dq ’ dr ’ 
or, what is the same thing, we have 
d 
d 
- J ( °’ 
dx’ 
dy’ 
(a, (b, H)) — (b, 
5 
h 
IT 
b), H), 
the identity in question. 
The Poisson-Jacobi Theorem. Art. Nos. 37 to 39. 
37. The foregoing identity shows that if (H, a) = 0, and (H, b) = 0, then also 
(H, (a, b)) = 0 ; or, what is the same thing, if a and b are solutions of the partial 
differential equation (H, 6) = 0, then also (a, b) is a solution ; or, say, if a, b are 
integrals, then also (a, b) is an integral.