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A MEMOIR ON DIFFERENTIAL EQUATIONS. 
[655 
and connected therewith the linear partial differential equation 
dHdO_dHdd dHdd_dHd6 + 
dp dx dx dp dq dy dy dq 
otherwise written 
where H denotes a given function of the variables: also the Hamiltonian system as 
augmented by an equality — dt, and as augmented by this and another equality 
G. Bipartite set (x, y, z, ... ; p, q, r,...). The partial differential equation 
H = const., where, as before, H is a given function of the variables, but p, q, r,... 
are now the differential coefficients in regard to x, y, z,... respectively of a function 
V of these variables, or, what is the same thing, there exists a function 
V= ( pdx + qdy + rdz +...), 
of the variables x, y, z,.... 
* 
In what precedes, I have written (x, y, z, ...) to denote a set of any number n 
of variables, and (x, y, z,...; p, q, r,...) to denote a set of any even number 2 n of 
variables, and the investigations are for the most part applicable to these general 
cases. But for greater clearness and facility of expression, I usually consider the case 
of a set (x, y, z, w), or (x, y, z\ p, q, r), &c., as the case may be, consisting of 
a definite number of variables. 
The greater part of the theory is not new, but I think that I have presented 
it in a more compact and intelligible form than has hitherto been done, and I have 
added some new results. 
Introductory Remarks. Art. Nos. 1 to 3. 
1. As already noticed, a letter not otherwise explained is considered as denoting 
a function of the variables of the set ; but when necessary we indicate the variables 
by a notation such as z = z (x, y)\ z is here a function (known or unknown as the 
case may be) of the variables x, y, the 2 on the right-hand side being in fact a 
functional symbol. And thus also z = z (x, y), = const, denotes that the function z (x, y) 
of the variables x, y has a constant value, which constant value is = z, viz. we thus 
indicate a relation between the variables x, y. 
2. The variables x, y, &c., may have infinitesimal increments dx, dy, &c. ; and 
the equations of connexion between the variables then give rise to linear relations 
between these increments, the coefficients therein being differential coefficients and,