100 
A MEMOIR ON DIFFERENTIAL EQUATIONS. 
[655 
16. There is no general process for obtaining an integral a of the differential 
equations. Supposing such integral known, we can introduce it as a variable, in 
place of one of the original variables, say w, viz. we thus reduce the system to 
dx _dy _dz _ da 
where X, Y, Z now denote the values assumed by these functions upon expressing 
therein w as a function of x, y, z, a, viz. they are now functions of x, y, z, a. The 
system thus breaks up into da = 0 and the system 
dx _dy _ dz 
X~~Y ~~Z' 
in which last (by virtue of the first equation, or a = const.) a is to be regarded as 
a constant; the original system of three equations between four variables is thus 
reduced to a system of two equations between three variables. Supposing b to be 
an integral of this reduced system, b is given as a function of x, y, z, a, but upon 
substituting herein for a its value as a function of x, y, z, w, we have b a function 
of the original variables x, y, z, w, and b is then a second integral of the original 
system. 
17. In like manner supposing a and b to be known, we reduce the system to 
the single equation 
dx dy 
X~T’ 
where X, Y are now functions of x, y, a, b; supposing an integral hereof to be c, 
we have c a function of x, y, a, b; but upon substituting herein for a, b their values 
as functions of x, y, z, w, we have c a function of x, y, z, w, and as such it is the 
third integral of the original system. 
18. It may be remarked that if, to the original system, we join on an equality 
-dt, viz. if we consider the system 
dx _ dy _dz _ dw , , . 
X —~Y~~Z ~~ If ’ 
where X, Y, Z, W are as before functions of the variables (x, y, z, w), then the 
integrals a, b, c of the original system being known, we can by means of them 
express for instance X as a function of x, a, b, c, and we have then, const. = t — 
dx 
where the integration is to be performed regarding a, b, c as 
constants; writing 
but after the integration replacing a, b, c by their values as functions of x, y, z, w, 
we have r a function of x, y, z, w; and we say that t — t is an integral; putting 
it = const, we use also t to denote the constant value of the 
question. Observe that here, the integrals a, b, c being known, the last integral t-r 
is obtained by a quadrature.