1888.] associated with Linear Differential Equations. 



315 



tt iv +6Ry + R^ = 0, 



to which the explicitly general qnartic can be reduced by the solution 

 of linear differential equations of the second and the third order 

 respectively. The differential equation satisfied by the quotient of 

 two solutions of the quartic is obtained ; and in this connexion there 

 arises a quartic quotient-derivative. Finally, the associate equations 

 of the quartic are formed ; and it is verified that all their invariants 

 are expressible in terms of the invariants of the original quartic. 



The seventh section is really a digression from the main subject of 

 the paper ; it is concerned with the special class of functions which 

 occur in the preceding section and are called quotient-derivatives. 

 The quotient-derivatives of lowest order are * 



s'" , 3s" 

 3 s" , 3s 



[».«] 



3 ' 



s iv , 4s"', 6/' 

 s v , 5s iv , 10s 7 " 



and so on; in these the differential coefficient of highest order which 

 occurs is of odd order, and thence these derivatives are said to be of 

 odd order. The two most important propositions which relate to them 

 are, first, if 



[ff, *]m = 0, [s, Z] n = 0, [z, X]p = 0, 



then [o-, z~\p = 0, 



where p — 1 = (m — 1) (« — 1) (p — 1) ; 



and second, that the law of change for homographic transformation of 

 both variables is 



ras + b ez+fl _ (ad-bc) n (gz + hy»' 



Lc6- + tZ' jr+*J« ~ (ek-fgr 3 lS ' Zjn ' 



There is then investigated the series of similar functions of even order 

 in the form 



s" , 2s' 



s' ' , 3s ' , 3s 

 s iT , 4s'", 6a" 



and so on ; and a connexion between the two classes is given. 



9o' 



