246 



Prof. Boole on Differential Equations, 



[May 26, 



is extremely small. It will, indeed, be a most remarkable conclusion, 

 should it ultimately prove that the primary solvable forms in question 

 stand in some absolute connexion with a certain class of algebraic equations. 



The following paper is a contribution to the general theory under the 

 aspect last mentioned. In endeavouring to solve Mr. Harley's equation by 

 definite integrals, I was led to perceive its relation to a more general equa- 

 tion, and to make this the subject of investigation. The results will be 

 presented in the following order : — 



First, I shall show that if u stand for the mth power of any root of the 

 algebraic equation 



n 91—1 1 [\ 



y -xy —1=0, 

 then w, considered as a function of x, will satisfy the differential equation 



in which e^=Xy D = -^, and the notation 

 dd 



(a_2) .. («— 5+1) 



is adopted. 



Secondly, I shall show that for particular values of in, the above equa- 

 tion admits of an immediate first integral, constituting a differential equation 

 of the w— 1th order, and that the results obtained by Mr. Harley are par- 

 ticular cases of this depressed equation, their difference of form arising from 

 difference of determination of the arbitrary constant. 



Thirdly, I shall solve the general differential equation by definite in- 

 tegrals. 



Fourthly, I shall determine the arbitrary constants of the solution so as 

 to express the mth power of that real root of the proposed algebraic equa- 

 tion which reduces to 1 when 07=0. 



The differential equation which forms the chief subject of these investi- 

 gations certainly occupies an important place, if not one of exclusive im- 

 portance, in the theory of that large class of differential equations of which 

 the type is expressed in (3). At present, I am not aware of the existence 

 of any differential equations of that particular type which admit of finite 

 solution at all otherwise than by an ultimate reduction to the form in 

 question, or by a resolution into linear equations of the first order. It 

 constitutes, in fact, a generalization of the form 



^+ D(D-l) ' : 



given in my memoir " On a General Method in Analysis " (Philosophical 

 Transactions for 1844, part 2). 



