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[656 
656. 
ON THE THEORY OF PARTIAL DIFFERENTIAL EQUATIONS. 
[From the Mathematische Annalen, t. xi. (1877), pp. 194—198.] 
In what follows, any letter not otherwise explained denotes a function of certain 
variables (x, y, p, q), or (x, y, z, p, q, r), &c., as will be stated in each particular case. 
An equation a = const, denotes that the function a of the variables is, in fact, a 
constant (viz. by such equation we establish a relation between the variables) : and when 
this is so, we use the same letter a to denote the constant value of the function in 
question; I find this a very convenient notation. 
Thus if the variables are x, y, z, p, q, r and if p, q, r are the differential coefficients 
in regard to x, y, z respectively of a function V of x, y, z, then H (as a letter not 
otherwise explained) denotes a function of x, y, z, p, q, r and considering it as a given 
function, 
H = const. 
will be a partial differential equation containing the constant H. For instance, if H 
denote the function pqr — xyz, H = const, is the partial differential equation, pqr — xyz = H 
(a given constant). 
The integration of the partial differential equation, H = const., depends upon that of 
the linear partial differential equation 
(H, ©) = 0, 
where as usual (H, 0) signifies 
0(H, 0) 3(H, 0) 9(H, 0) 
d(p, x) + d(q, y) + d(r, z) 
It can be effected if we know two conjugate solutions a, b of the equation (H, ©) = 0, 
viz. a, b as solutions are such that (H, a) = 0, (H, b) = 0, and (as conjugate solutions) 
are also such that (a, b) = 0 ; in this case if from the equations 
H = const., a = const., b = const.