378 
(£)-(B)-»(f)-» 
and, by eliminating a, 8, y between the four equations stated, we would 
get a partial differential equation in three independent variables, in ge- 
neral non-linear, 
dw dw dw 
F(s, Y; 2, W, daz’ dy’ 7 = 0, 
of which (II) would be said to be a complete primitive. 
If, instead of differentiating with respect to the independent vari- 
ables, we had differentiated with respect to the arbitrary constants, thus 
getting 
Oo ee oes 
da ’dBp ’dy 
and then eliminated the arbitrary constants between the given equation 
and these three, we would obtain a result of the form 
0, 
F (a, y, s, w) =0, 
which will, in general, satisfy the partial differential equation, previously 
derived, 
dw dw dw 
E(2, 48,0 7 Go = \=0, 
and, as it exhibits no arbitrary constants, may be denominated its sin- 
gular solution. 
4, Again if, w and v being any functions of known form in 2, y, 3, w, 
we were given an equation of the type 
Siu, Y, 3, Ww, d(u, v)} = 0, (I1') 
@ being any arbitrary function, we may differentiate with respect to the 
independent variables 2, y, s, and eliminate 
dp dp 
9; Wa’. dy’ 
thus again obtaining a partial differential equation 
F ; dw dw ) z 
“ x,y; » W, dx’ dy’ dz ales td 
of which (II’) may be said to be the general primitive. 
5. Proceeding by analogy, being given any equation in two indepen- 
dent variables, but exhibiting five arbitrary constants, 
S (2, Y, %, a, B, 1, 6, e) = 0, (IIT) 
we may differentiate this equation twice successively with respect to x 
