655] 
A MEMOIR ON DIFFERENTIAL EQUATIONS. 
107 
may be of a more special form; for instance, it may be a sum of two terms = %dp + yda: 
or, finally, it may be a single term =i~dp, and in this case we have the Hamiltonian 
system, viz. writing H in place of p, if we have 
Xdx +Ydy + Zdz + Pdp + Qdq + Rdr = %dH, 
where H is a given function of the variables, then the system is 
dx _ dy _ dz _ dp _ dq dr 
dH dH dH _dU _dH _dJT 
dp dq dr dx dy dz 
which is the system in question. 
33. Any integral a of the system is a solution of 
dH de dH dd + dH dd _dH dd _dH dd _dH d£ 
dp dx dq dy dr dz dx dp dy dq dz dr 
viz. writing, as above, 
(tt a\ d (H, e) d(H, e) d(H, e) 
K d{p, x) + d(q, y) + d(r, z) 
= last-mentioned expression, 
the partial differential equation is (H, d) — 0 ; and, conversely, any solution of this 
equation is an integral of the differential equations. 
34. It is obvious that a solution of (H, d) = 0 is H ; hence the entire system of 
independent solutions may be taken to be H, a, b, c, d; or, if we choose to consider 
a set of five independent solutions a, b, c, d, e, then we have H = H(a, b, c, d, e) a 
function of these solutions. 
An Identity in regard to the Functions (H, d). Art. Nos. 35 and 36. 
35. Taking the variables to be (x, y, z, p, q, r), and H, a, b to be any functions 
of these variables, we have the identity 
(H, {a, b)) + (a, (b, H)) + (b, (H, a))=0, 
which is now to be proved. For this purpose we write it in the slightly different form 
((a, 6), H) = (a, (b, H))-(b, (a, H)). 
The first term on the right-hand side is 
'da d da d da d 
jZ "h ZJ7. 4" ZT. 
) 
operating upon 
db dH db dH db dH db dH db dH db dH