[655 
655] . A MEMOIR ON DIFFERENTIAL EQUATIONS. 99' 
Observe that, in speaking of an integral a, we mean a function of the variables; 
the differential equations give between the variables the relation a = const., and when 
this is so, we use the same letter a to denote the constant value of this function. 
h. between 
3; in fact, 
space, and 
tios of the 
itive point; 
e equations 
The locus 
a threefold 
the system 
volve three 
,s functions 
ats will be 
respectively 
12. In speaking of the three integrals a, b, c we mean independent integrals; 
any function whatever (ft a of an integral a, or any function whatever (ft (a, b) of two 
integrals a, b, is an integral (viz. it is an integral of the differential equations, and 
also a solution of the partial differential equation), but such dependent integrals give 
nothing new, and we require a third independent integral c, viz. we need this to 
express the threefold relation between the variables, given by the differential equations, 
and also to express the general solution eft (a, b, c) of the partial differential equation. 
13. By what precedes the analytical condition, in order that the integrals a, b, c 
may be independent, is that they are such as not to satisfy the relations 
d (a, b, c) _ 
d(x, y, z, w) 
14. We moreover see d posteriori, that there cannot be more than three inde 
pendent integrals; in fact, if a, b, c, d are integrals, then, considering them as 
solutions of the partial differential equation, we have four equations which by the 
elimination therefrom of X, V, Z, W, give 
d {a, b, c, d) _ 
d(ao, y, z, w) 
increment 
and this is the very equation which expresses that a, b, c, d are not independent. 
15. The notion of the integrals may be arrived at somewhat differently thus: 
take a, b, c, d any functions of the variables, and write 
r, Y, Z, W 
z. that we 
a y da da „ da jrr da 
-A — A -j (-.* j Y Zj —[- W -=— 
dx dy dz dw 
and the like for B, G, D; then replacing the original variables x, y, z, w by the 
new variables a, b, c, d, the differential equations become 
ition. 
da db dc dd 
an integral 
equation is 
1 equations 
in general 
and c are 
A ~ B 
where A, B, C, D are to be (by means of the given values of a, b, c, d as functions 
of x, y, z, w) expressed as functions of a, b, c, d. If then A = 0, B— 0, (7=0, the 
differential equations become 
da db dc dd 
W~o “ o"~D ; 
itial equation, 
regard to the 
viz. we have da = 0, db = 0, dc = 0, and therefore a = const., b = const., c = const., that 
is, we have the integrals a, b, c as before. 
13—2