655] 
93 
655. 
A MEMOIR ON DIFFERENTIAL EQUATIONS. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xrv. (1877), 
pp. 292—339.] 
We have to do with a set of variables, which is either unipartite (x, y, z, ...), 
or else bipartite (x, y, z, p, q, r, ...), the variables in the latter case corresponding 
in pairs x and p, y and q, &c. 
A letter not otherwise explained denotes a function of the variables. Any such 
letter may be put = const., viz. we thereby establish a relation between the variables; 
and when this is so, we use the same letter to denote the constant value of the 
function. Thus the set being (x, y, z\ p, q, r), H may denote a given function 
pqr — xyz ; and then, if H = const., we have pqr — xyz =H (a constant). This notation, 
when once clearly understood, is I think a very convenient one. 
The present memoir relates chiefly to the following subjects: 
A. Unipartite set (x, y, z,...). The differential system 
dx _dy _dz _ 
X~T~ Z ~ ’ 
and connected therewith the linear partial differential equation 
also the lineo-dififerential 
v I ~\7 I r/ dO 
A-j—f-r -7——h 
dx dy dz 
= 0: 
Xdx + Ydy + Zdz + .... 
B. Bipartite set (x, y, z,p, q, r,...). The Hamiltonian system 
dx 
dy 
dz 
dp 
dq 
dr 
dH~ 
~ dH~ 
~dH~‘" 
' ~ dH 
dH 
dH 
dp 
dq 
dr 
dx 
dy 
dz