278] NOTE ON THE SINGULAR SOLUTIONS OF DIFFERENTIAL EQUATIONS. 427 
and substituting these values in the function F(P, Q,... P', Q',...) it is clear that we 
shall have F(P, Q,..., P', Q',...) = UX'Y'Z'... where U is a symmetrical function of 
X, Y, Z, &c., and therefore a function of P, Q,... ; and this equation will be identically 
true whatever values we attribute to X', Y', Z',..., hence putting these quantities 
respectively equal to unity, we have 
P' = n, 
Q'=(n-l)P, 
R'=(n- 2) Q, 
&c., 
and with these values 
U=F(P, Q,...,P, Q',...), 
that is U = 0 is the result obtained by eliminating c from the primitive equation and 
the equation 
wc n-1 + (n — 1) Pc n ~ 2 + ... =0, 
which is the equation obtained by differentiating the primitive equation with respect 
to the arbitrary constant c: that is, U= 0 being the singular solution, the differential 
equation is 
UX'Y'Z'... = 0. 
It is to be remarked that (P, Q, &c. being rational and integral functions) then 
if the roots X, Y, Z, &c. are also rational and integral functions, the differential 
equation contains 17 as a separable rational and integral factor, but if the roots are 
irrational then the differential equation does not really contain the rational and integral 
factor U, but X'Y'Z'... is here a rational fraction containing U in the denominator 
and UX'Y'Z'... is an indecomposable rational and integral function. This is easily 
verified a posteriori for a quadratic equation. 
2, Stone Buildings, W.C., 28th January, 1858.