201 



is known to lead directly to the solution of the cubic resol- 

 vent. If we change the independent variable by assuming 



X-=: 



we are conducted to the equation 



Sin t dv' dt^ 



sm «f ^ ^ dt 



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. 25th, 1862, 

 (of which he kindly permits me to make this use), remarks, 

 ^' I see how it could be solved by a definite integral; but that 

 is not what we want, 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 new primary form, 

 and then it constitutes a real addition to the theory of 

 diiFerential equations. So also," he adds, "for the quintic 

 resolvent, which no doubt is a new 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, but it now seems that 

 I had not." These remarks, from one who has contributed so 

 largely to the theory of differential equations, will no doubt 

 be read with pleasure by all who 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," 





