ON THE THEORY OF THE SINGULAR SOLUTIONS OF DIFFER 
ENTIAL EQUATIONS OF THE FIRST ORDER. 
[From the Messenger of Mathematics, vol. n. (1873), pp. 6—12.] 
I CONSIDER a differential equation under the form 
0> y> p) = 0, 
where 
1°. <f) is as to p, rational and integral of the degree n; 
2°. it is, or is taken to be, one-valued in regard to (x, y); 
3°. it has no mere (x, y) factor; 
4°. it is indecomposable as regards p. 
Considering (x, y) as the coordinates of a point in piano, the differential equation 
determines a system of curves, in general indecomposable, the system depending on 
a single variable parameter, and such that through each point of the plane there pass 
n curves. 
Such a system is represented by an integral equation 
f(x, y, c 1} c a ...c m ) = 0, 
where 
5°. / is rational and integral in regard to the m constants, which constants are 
connected by an algebraic (m — 1) fold relation; 
6°. it is, or is taken to be, one-valued in regard to (x, y); 
7°. it has no mere (x, y) factor; 
8°. it is indecomposable as regards (x, y); 
C. VIII. 
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