OF LINEAR DIFFERENTIAL EQUATIONS. 245 



every differential resolvent of an order higher than the 

 second ^ gives us, at least when the dexter of its defining 

 equation vanishesf, a new primary form_, that is to say, a 

 form not recognized as primary in Professor Boole^s theory. 

 And in certain cases in which the dexter does not vanish, 

 a comparatively easy transformation will rid the equation 

 of the dexter term, and the resulting differential equation 

 will be of a new primary form. The same transformation 

 which deprives the algebraic equation of its second term 

 will deprive the differential equation of its dexter term. 

 Thus [ex. gr.) if we write ^ + i in place of 2/, the equation 

 (II) becomes 



(r+ i)^—w(^+ 1)"-' + (w— 1)^=0, 



and the resolvent (C) becomes 



[/iB— 2]"-V(^+ i)= [?i— i]»-V, 

 which, since 



{n—i)[riD—n—i)[nD — 2]''-^^ 

 = {n—i){n-'n—i)[n — 2Y'-^e^ 

 = —{n—i)\n — 2Y-''^ 

 = — [ti— i]«-'e^, 



may be written simply 



n''-^\_(n—i)'I)Y-''z—[n—i){nT>—n—i) 



* The resolvent of the trinomial cubic of the form (I) has long been 

 known as solvable. This resolvent is of course of the second order. 



t The qualification in the text is necessary, because, of the two equations 



^(D)y=X, 0(D)y=o, 



the solution of the former does not in general enable us to obtain that of the 

 latter, though from that of the latter it is well known that the solution of 

 the former can be obtained. 



