152 Proceedings of the Royal Society of Edinburgh. [Sess. 
possessing a singular solution may often be considerably simplified in form 
by the substitution 
Z = singular solution, 
and thus facilitate its integration. 
Among the various cases that may occur in the derivation of the 
singular solution of <p{xyzpq) = 0, there is one of special importance, viz. 
when the expressions for p and q derived from 
* render 
<f> p = 0 and <f> q = 0 
<£= 0 , 
i.e. when the three equations are equivalent to only two independent 
equations. 
In this case, if p and q satisfy the symbolical relation 
the integral of the equation 
p q -l 
_0 d_ d_ 
dx 3 y dz 
p q -l 
pdx + qdy - dz = 0 
(2) 
will furnish a onefold of singular solutions. On referring to equation (1), 
it is at once evident that this corresponds to the case where D = 0, for then 
the derivation of the singular integral leads to 
P fxjfzi q fy/f v> 
and these satisfy the three conditions 
while the integral of 
is 
<£ = o, <f> p = o, 4> q = o, 
pdx + qdy - dz — 0 
f(xyz) = c. 
Hence the general Partial Differential Equation of the first order which 
has the given onefold of Singular Solutions f(xyz ) = c is 
(/>= A(p +f x /f z y + 2B (p +f x /f z ) (q +fylf z ) + c (q + A) 2 = 0 . . (3) 
where A, B, and C are arbitrary functions of x, y, z, p and q such that is 
irreducible as regards p and q, and is a synectic function of its variables. 
The following method of constructing such equations was given by the 
late Professor Chrystal in his class lectures. 
Commencing with an equation <p(xyzpq) = 0, having a singular solution 
x(xyz) — 0, then (p(xy zp q) = x( x V z ) will become an identity when the values 
