﻿362 Sir James Cockle on Fractional Criticoids. 



e_ of—ce __ be— of 

 *-~a~c~^b» V -"a7=b*' 



0=4>&n)-*-fn-ff ( 2 ) 



Consequently 



4>(® + & V + V) = <*> x * + 2bxy + cz/ 2 + e% +fri +g, 

 and the transformed quotic is therefore 



^ + Zbxy + Cf+ e(bf-ce)+f{be-af) ^ g . 



uC 



whereof the absolute term, or critical function, may be equated 

 with either side of the identity, 



e (bf- ce) +f(be -af)+g (ac - 6 2 ) 2bef- af 2 - ce* 



ac 



-b* ac-b* + 9 ' 



4. When the equations, whereof any two entail the third, 



a - - - e (K 



~b-c~f> (3) 



are, one or more of them, satisfied, we have particular cases of 

 the transformation. If all are satisfied, the given quotic may be 

 written 



1 e 



-{ax + by) 2 + 2- [ax + by)+g. 

 a a 



In such a case fi is a multiple of X, the equations \=0 and 

 ft =0 are no longer independent, the quotic in substance involves 

 only one variable, and the critical function is 



-(aZ + brtf-zUat + brD+g. 



U (I 



If we use the symbol k as a general representation of critical 

 functions, and reduce the formula last preceding by means of 

 \=0 or fJb = 0, we have 



j. e * 



u a 

 which, in virtue of the relations 



a b 



may be written in various forms. When ac~b' 2 =0, and e = 

 and/=0,we have a still more restricted quotic. Whence— Z> 2 =0, 

 and the other equations of the group (3) are not satisfied, then 

 there is no critical transformation. When a, b, and e, or when 

 b } c, and /vanish, we again have restricted quotics. 



5. When be — af=0, and the other equations of the group (3) 



