201 
is known to lead directly to the solution of the cubic resol- 
vent. If we change the independent variable by assuming 
1 
Xzx. 
o 
cos 3 t, 
we are conducted to the equation 
cos 3 1 d 3 y 
2 2 .3 3 
sm t 
— 2 3 .3 3 cos 2 t.f* 
dt 3 d(- 
— (29—32 sin 2 t) . 
sin < v 1 dt 
— 5y=0 
the complete integration of which will of course give the roots 
of the biquadratic. 
In reference to the symbolical form of the biquadratic 
resolvent given in my last communication to the Society, 
Dr. Boole, in a letter to myself, under date Feb. 25tli, 1862 , 
(of which he kindly permits me to make this use), remarks, 
“ I see how it could he solved by a definite integral; hut that 
is not what we Avant, I presume. If it do not admit of 
resolution or reduction to forms recognized as primary in my 
theory, it must itself be considered as a neAV primary form, 
and then it constitutes a real addition to the theory of 
differential equations. So also,” he adds, “for the quintic 
resolvent, Avhich no doubt is a neAV form. If you have my 
Finite Differences, I would ask you to look at the conclusion 
of Chapter IX. I certainly thought that I had found all the 
primary forms for binomial equations, hut it now seems that 
I had not.” These remarks, from one Avho has contributed so 
largely to the theory of differential equations, will no doubt 
be read with pleasure by all Avho are interested in the pro- 
gress of Algebra or the cultivation of the Calculus. It will, 
certainly, as Dr. Boole in a more recent letter to me (dated 
March 1st) observes, “Be a remarkable result if it should 
ultimately prove that the primary forms of integrable, linear, 
differential equations, stand in some close connexion with the 
solvable forms of algebraic equations.” 
