99 
9. Lagrange, in fact (ib. pp. 182—183), observes that 
since the derived equation Y(a) = 0 contains the condition 
which renders two of the roots of F(a?, y , a) = 0 equal, conse- 
quently the singular value of y gives to the last equation a 
double root. 
10. Lagrange also remarks (ib. p. 219), that the singular 
primitive of f(x, y, y) = 0 is obtained by the aid of the two 
equations /(/) = 0, f(%, y) = 0 and the elimination of y ' by 
means of the proposed equation. He adds that if the two 
results give the same relation between x and y that rela- 
tion will be the singular primitive ; if not, it will be a sign 
that the proposed equation has no such primitive. 
11. But f'(y) = 0 is the condition of the equality of two 
of the roots of the given differential equation, treated as an 
equation in y\ 
12. Mr. (now Prof) Cayley long ago stated that the 
singular solution is the result of the elimination of the 
arbitrary constant between the primitive and its derived 
equation with respect to the constant (Q. J. of Math., 1860, 
vol. III., p. 36 ; see also the Messenger of Math., N. S., 
complete primitive involving an arbitrary quantity of two or more dimen- 
sions or that each of the roots requires a multiplier. 
I take this opportunity of observing that in a letter to me, dated 
August 20th, 1862, Mr. Eawson remarks that an equation deduced from 
a cubic by one differentiation can be made linear by a proper assumption. 
Such a linear equation I call a first linear resolvent. 
I also remark that I have verified one of the results of Mr. Eawson’ s 
recent paper [ante, p. 59). We may infer from what I proved years ago 
(see Phil. Mag. for May, 1861), that every cubic in y has a second linear 
resolvent of the form y" +vy' + y,y— M. 
Hence, slightly generalizing the process given in my Second Chapter 
on Coresolvents (Q. J. of Math., vol. VI., p, 226), we get 
2y" +v'2y'+ ^y—S M 
or, putting ~ 2 y—a, 
a" + vd/ + ya—SM. 
and M vanishes with a. Next I find 
v= — ^log(r u' — uv') 
and, summing again, I, after easy reductions, verify Mr. Eawson’s 
logarithmic coefficient. I have not calculated p, but the process is 
applicable to p. 
