426 
[278 
278. 
NOTE ON THE SINGULAR SOLUTIONS OF DIFFERENTIAL 
EQUATIONS. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. III. (1860), 
pp. 36, 37.] 
The following investigation (which has been in my possession for a good many 
years) affords I think a simple explanation of the theory of the singular solutions of 
differential equations. 
Let the primitive equation be 
c n + p c n-i + Q c n-2 + " = o, 
where c is the arbitrary constant and P, Q... are any functions of x, y ; then the 
differential equation is obtained by eliminating c from the foregoing equation and the 
derived equation 
P'c"“ 1 + Q'c n ~ 2 + ... = 0, 
and the result may be represented by 
F(P, Q,..., P', Q\...) = 0. 
Assume now 
c 11 + Pc 11-1 + Qc n ~ 2 + ... =(c + X)(c + Y)(c + Z)..., 
then we have 
P=X+Y+Z+ &c., 
Q=XY+XZ+ YZ+&L e., 
&c., 
and consequently 
P' =X' + Y' + Z' + &c., 
Q' = (Y + Z + &c.) X' + &c., 
&c.,