176 On Differenticd Eqiicdions 



Art. YI. — On Differential Equations and on Co-resolvents. 

 By the Hoxourable Chief Justice Cockle, F.R.S., 

 President of the Queensland Philosophical Society, &c. 

 Communicated by the Honourable Sir Redmond Barry, 

 Chancellor of the University of Melbourne, &c. 



[Read 11th June, 1866.] 



§ 1. On Differential Equations. 



1. I propose to show that any linear differential equation 

 whatever can be deprived of its second and third terms 

 simultaneously, provided that we are at liberty to assume 

 the solution of the general linear differential of the second 

 order ; or, in other words, that the annihilation of the 

 second and third terms of any linear differential equation 

 may be made to depend upon the solution of a linear differ- 

 ential equation of the second order. This is the analogue of 

 the proposition that the second and third terms of an alge- 

 braical equation may be made to vanish simultaneously by 

 means of the solution of a quadratic only. 



2. Since any linear differential equation can be deprived 

 of its second term by solving a linear differential equation 

 of the first order only, we are at liberty to start from the 

 equation of the third order — 



fl^ + ^rjy + sy = - -(a) 

 dx^ dx ^ ^ ^ 



3. Now, inasmuch as the complexity of the formulae ren- 

 ders it necessary, or at all events desirable, to abridge the 

 notation as much as possible, I shall have recourse to the 

 following abbreviations. I shall denote differentiations 

 with respect to the variable x by acute accents ('), and 

 differentiations with respect to a new independent variable 

 t by grave accents ('). Thus I shall wiite — 



^ = y, p, = y", pi = r. &c. 



dx ^ ' dx^ ^ ' dx^ ^ ' 



dt - y^ dt^ - y ' dt^ - y ' ^^' 



The letters r and s in equation (a) denote any constants or 

 any functions of x, the multiplier ' 3 ' being prefixed to r 



