106 
A MEMOIR ON DIFFERENTIAL EQUATIONS. 
[655 
30. According as 
Pfaffian Theorem. Art. No. 30. 
the variables are 
we have 
x, 
X, y, 
x, y, z, 
X, y, z, w, 
Xdx 
= du, 
Xdx 4- Ydy = \du, 
Xdx + Ydy + Zdz = \du + dv, 
Xdx 4- Ydy + Zdz 4- Wdw = \du 4- pdv, 
and so on; viz. the theorem is that, taking for instance two variables, a given lineo- 
d inferential Xdx + Ydy is = \du, that is, there exist A, u functions of x, y, which verify 
this identity, or, what is the same thing, such that we have 
X, Y=\ 
du 
dx ’ 
A 
du 
dy ] 
and so, in the case of three variables, there exist A, u, v functions of x, y, z, such that 
X, Y, Z= A 
du 
dx 
dv 
dx’ 
du dv 
dy dy ’ 
dv 
dz‘ 
The problem of determining the functions on the right-hand side is known as the 
Pfaffian Problem; this I do not at present consider, but only assume that there exist 
such functions. 
The Hamiltonian System, its derivation from the general System. Art. Nos. 31 to 34. 
31. Considering a bipartite set (x, y, z: p, q, r), the general system of differential 
equations may be written 
dx _dy _dz _ dp _ dq dr 
P-~Q-R--X = - Y = -£• 
But by the Pfaffian theorem we may write 
Xdx + Ydy + Zdz 4- Pdp 4- Qdq 4- Rdr — %dp + rjda 4- %dr, 
viz. there exist f, ij, £ p, a, t functions of the variables x, y, z, p, q, r, such that we 
have 
v c.dp der dr 
^ dx ^ dx ^ dx' 
P =f ^ + „^ + 1-± 
^ dp ^ dp ’ dp' 
and we have the foregoing general system expressed by means of these given functions 
£, Vi P> a > T of the variables. 
32. But the lineo-differential 
Xdx + Ydy 4- Zdz 4- Pdp + Qdq 4- Rdr