DKIUVATIVES ASSOCIATED WITH l.INHAR DIFr'KKKNTIAL EQUATIONS. 445 



gives 



0= -2{<r, z}, i.e., {<r, z] = 0. 



As we are seeking the values of special integrals, it will suffice to obtain them in as 

 simple a form as possible. Since <r may not be a constant, we therefore take a- = z, 

 and the corresponding values of u are 



and consequently 6 = 0, which agrees with the former result.* In this case the 

 quotient-equation is 



S T , 5**, 10s" 1 

 s*, 4s lu , 6s" 

 ", 3s", 3s 1 



= 



(25), 



and the primitive of this equation is 



s = 



A + Rg + Cc* 



A' + Wz + CTc 



(26). 



The function on the left-hand side of (25) will be called the quotient-derivative 

 associated with the cubic, or, more shortly, the cubic quotient-derivative ; the corre- 

 sponding function for the equation of the second order, viz., 



s" 1 , 3s" 

 s", 2s 1 



which is 2s' 2 {s, z}, being a multiple of the Schwarzian derivative, may be called the 

 quadratic quotient-derivative. The consideration of these derivatives, and of others 

 of higher order, will be resumed later ; but it may be mentioned that, if denote the 

 Schwarzian derivative {s, z}, a (= 2s' 2 ff), and T 3 respectively the quadratic and the 

 cubic quotient-derivatives, then 



and the equation (24) is 



- 27s' 3 e 2 - 



" 2 - 54s'V + OOs'sV"). 



88. A particular case referred to by MAi,ETt is at once reducible to one of the cases 

 considered in 83 ; for, supposing s = (az + b)/(cz -f- d) so that 6 vanishes, we have 



and, therefore, 



's ul = 3s"*, 

 = 2s u s m . 



* The result would similarly follow, if a were taken in its general form (at + 6)/(c* + d). 

 t ' Phil. Trans.,' 1882, p. 759. 



