1911-12.] Singular Solutions of Partial Differential Equations. 151 
Any integral which satisfies these latter two conditions, we call a 
singular solution. 
If such a solution therefore exists, we may obtain it by the elimination 
of p and q between the three equations 
(a) </> = 0, (6) <f> p = 0, (c) <f> q = 0. 
There are two fundamentally distinct methods by which the problem 
of the singular solution may be attacked. In the first place, we may treat 
the question analytically by discussing the particular forms of singularity 
that may exist in the general equation (p(xyzpq) = 0 ; or, secondly, we may 
apply a synthetic method, viz. given the singularities which are to occur, 
construct the general differential equation which possesses them, imposing 
in the latter case such restrictions upon the function as to ensure that it 
admits of an integral. 
In the earlier part of this paper I propose to treat this problem from 
the synthetic point of view. 
Darboux in his “ Memoire sur les Solutions Singulieres des Equations 
aux Derivees Partielles du Premier Ordre ” (Memoires presentes a V Academie 
des Sciences , tome xxvii.), gives the following differential equation as the 
general form of an equation possessing the singular solution z = 0 
2 4>(xy)z + A p 2 + 2B pq + C q 2 + D pz + E qz + F z 2 = 0, 
A, B, C, D, E, F being arbitrary functions of x, y, z, p, and q. 
Although this is apparently true merely for the particular case where 
the plane of x, y is the singular solution, yet the more general case where 
the surface f(xyz) = 0 is the singular integral can be immediately deduced 
from the above by the substitution 
z=f(xyZ), 
giving 
v =/« +fZp\ <i =f y 
where 
, 0Z , 0Z 
p= Tx’ q= ^’ 
and omitting the dashes and writing 0 for Z we finally deduce that the 
differential equation of the first order, having the singular solution 
f(xyz) — 0, is given by 
<£= A(p +f x /f z ) 2 + 2B (p +f x /f z ) (q +f y ff z ) + C(q +f y /f z ) 2 + 1 V(xyz) = 0 . (1) 
where A, B, C, D are arbitrary functions of x, y , 2 , p and q, such that <p is 
an irreducible and synectic function of these arguments. 
This latter transformation suggests at once that a differential equation 
