383 
tion in three independent variables (and a corresponding method will 
apply generally) 
let us assume for w the linear form 
wai +e 42 ie 
nd aa ae 
where a, B, y, « are arbitrary constants. Then, by substitution, we find 
that this will be the solution required, if the arbitrary constants be con- 
nected by the relation 
a+B+y=1. 
Hence the complete primitive of the given equation is, writing y for «, 
a a ee 
ae, botanpy. 
and the general primary 
pas ee + (a, B) | 
a B* i-(@+8) 
PEON Wo ee en ten 
one oaks th A) 
d(x y g 
0" late mer R om 
T have ventured to denominate the system of equations just written, the 
‘general primary’ solution of the given partial differential equation, in- 
stead of, as it is usually denominated, the ‘ general primitive.’ It would 
seem to be right to distinguish between the two cases, where one arbi- 
trary constant is made an arbitrary function of one or more others, and 
where an arbitrary function of the variables is introduced. It is true, 
indeed, that in some cases these duplicate solutions coincide, as, for in- 
stance, in the case of the general functional equation of surfaces of revo- 
lution, namely, 
la + my +ns= (2 +? + 2) 
which may be readily identified with the general primary, obtained from 
the complete primitive 
(a —¢ cos)? + (y —¢ cos pm)? + (2 - ¢ cos)? = 9, 
by putting c = ¥(r); but such identification does not appear to be gene- 
rally possible. 
