150 
Proceedings of the Royal Society of Edinburgh. [Sess. 
XIII. — “ On the Singular Solutions of Partial Differential Equations 
of the First Order.” By H. Levy, M.A., B.Sc., Carnegie Scholar. 
(' Communicated by D. Gibb, M.A., B.Sc.) 
(MS. received March 4, 1912. Read May 6, 1912.) 
Section I. — Synthetic Treatment. 
The complete integral of the differential equation 
<f>(xyzpq) = 0 
is a relation among the variables, which includes as many arbitrary 
constants as there are independent variables. But it is important to 
distinguish carefully between differential equations which have been 
formed by the elimination of constants from some complete primitive, 
and those whose origin is quite unknown, or which may have been 
constructed by some method totally different from the first. 
In the original case, the differential equation can always be integrated 
in finite terms, while in the latter, only under the conditions laid down in 
Cauchy’s Existence Theorem can an integral be obtained, and even then 
usually as an infinite series. 
These conditions demand that <p(x yzpq) = 0 be an irreducible integral 
function of p and q , and a synectic function of all its arguments. Let 
z = <f)(xy) be any integral, regular in the neighbourhood of (x 0 y 0 ), and let 
z 0 , p Q , q 0 be the values of 0 , p, and q corresponding to x = x 0 , y = y 0 , then if 
P = d<p/dp> Q = d(p/dq, and P 0 4=0, the differential equation can be put in 
the form 
p = x( x v Z( i) 
where x is synectic near x 0 , y 0 , z 0 , and such that 
Po ~ x( x oyo z o^ t o)' 
It follows by Cauchy’s Existence Theorem * that the equation has 
one integral and only one, which reduces to 
z = <f>(x 0 y ) when x = x Q . 
Hence the supposed integral will always be furnished by Cauchy’s 
Existence Theorem, unless it involves 
P=0, Q = 0. 
* Goursat’s “ Le§ons sur l’integration des Eqvations aux derivees partielles. : 
