1911-12.] Singular Solutions of Partial Differential Equations. 153 
of p and q derived from it by differentiating partially with respect to p 
and q be substituted, so that this last equation will have a onefold of 
singular solutions. 
To a certain extent this was working in the dark. It was necessary, 
in the first place, to obtain a function 0 having a definite singular solution, 
either by guessing one, or by eliminating two arbitrary parameters from 
an equation having a Lagrangian Singular Solution, in either case usually 
entailing a great deal of labour. In the second place, it was extremely 
difficult by this method to work to an equation possessing the particular 
singular onefold that might be desired. On considering equation (1) it is 
seen that had the term associated with D been any other than f(xyz), there 
would have been no singular solution, nevertheless on putting D = 0, the 
resulting equation would still possess a singular onefold. But Professor 
Chrystal’s process of putting the singular solution to the right-hand side 
of the equation is really equivalent to putting D = 0. Hence, for the 
success of his method, we need not necessarily commence with an equation 
possessing a definite singular solution, but one which merely determines p 
and q by partial differentiation, to satisfy the symbolical relation (2). 
The following examples will serve to illustrate the foregoing : — 
Example 1.— Suppose f(xyz) — c is given by x^ + y 1 — cz^ = 1, a system 
of conicoids. 
• fjfz = -- ^/(^ 2 + y 2 - 1 )> fytfz = - W(^* 2 + y 2 - 1 )• 
Hence an equation with the required onefold would be 
which gives 
( px + qy — z ) 2 - p 2 - q 2 = 
-z 2 
x 2 + y 2 — 1 
We shall return to this example later, as it is one given by Professor 
Chrystal as illustrative of an equation possessing a onefold. 
Example 2. — Let the onefold be given by 
z = ax + (3y + c 
where a and /3 are constants, and c is arbitrary. 
Hence 0 may be written as 
2 A (p - a) r + 2B (q - py + 2C (p - a )*(q - = 0 . . . (4) 
where 
r and s are > 1, 
