636] 
19 
636. 
ON THE THEORY OF THE SINGULAR SOLUTIONS OF DIFFER 
ENTIAL EQUATIONS OF THE FIRST ORDER. 
[From the Messenger of Mathematics, vol. vi. (1877), pp. 23—27.] 
In continuation of the former paper with this title (Messenger, vol. ii., 1873, pp. 6—12, 
[545]), I propose to discuss various particular examples, chiefly of cases in which the 
differential equation is of the form (L, M, Nffp, 1) 2 = 0, where L, M, N are rational and 
integral functions of (x, y), and whether it admits or does not admit of an integral 
equation (P, Q, R][c, 1) 2 = 0, where P, Q, R are rational and integral functions of (x, y). 
The singular solution of the differential equation 
(L, M, NQp, 1) 2 = 0, 
if there be a singular solution, is S = 0, where S is either = LN — M 2 , or a factor of 
LN — iff 2 . But in general LN — M 2 is an indecomposable function, such that LN—M 2 — 0 
is not a solution of the differential equation, and this being so, there is no singular 
solution; viz. a differential equation (L, M, NQp, 1) 2 = 0, where L, M, N are rational 
and integral functions of (x, y), has not in general any singular solution. 
Consider now a system of algebraical curves U = 0, where U is as regards (x, y) 
a rational and integral function of the order m, and depends in any manner on an 
arbitrary parameter C *. I say that there is always a proper envelope, which envelope 
is the singular solution of the differential equation obtained by the elimination of C 
from the equation U= 0, and the derived equation in regard to (x, y). It follows 
that the differential equation (L, M, Nfp, 1) 2 = 0, which has no singular solution, does 
not admit of an integral of the form in question U= 0, viz. an integral representing a 
system of algebraic curves. 
* The expressions in the text may be understood as extending to the ease where U is a function of any 
number (a) of constants c 1# c 2 ,...,c a , connected by an (a-l)fold relation, U thus virtually depending on a 
single arbitrary parameter. 
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