﻿Sir James Cockle on Fractional Criticoids. 363 



are not satisfied*, we have 



£=--, „=0, k=g--=g- p 



and when bf—ce=0, and the other equations of the group (3) 

 are not satisfied, we have 



In the first case we have «f+e = 0, or bi;-\-f=Q; and in the 

 second we have cr) +f=0, or 6?7 + e = 0. All four forms are 

 embraced in the group (1). When the general critical function 

 becomes a vanishing fraction, we see that the limiting value of 

 the fraction gives the particular critical function. For the com- 

 parison by the above process of the critical functions of quotics, 

 it is of course necessary that the coefficients of the leading terms, 

 or terms of highest degree, in one should be equal to, or the 

 same multiple of, the corresponding coefficients in the other. 



6. In order to pursue the analogy between quotics and quo- 

 toids, let 



y]r(z) = ar + 2bs + ct + 2ep + 2fq +gz, 



wherein p, q, r, s, t have the meaning usually assigned them in 

 the theory of linear biordinals, and the coefficients a, b, c, e,f, g 

 are in general functions of the independent variables x and y. 

 Also let the differential coefficients of a certain auxiliary quan- 

 tity f be represented as follows : 



dx*~ P > dxdy~ <Ti dy*~ T ' dx~* dy~ V '' 



and form the expression e~ty(e£z) ; that is to say, substitute e^z 

 for z in the quotoid and divide the result by e^. The result, so 

 divided, will be 



ar + 2bs + ct + 2\p + 2nq + 6z, 



* When be-af=0, then 



and 



and when bf—ce=0, then 



and 



2bef-af 2 -ce 2 =(b 2 -ac)~ = (6 2 -«c)^, 

 a o 



bf -ce = {b 2 -ac) - = (b 2 -ac) {; 

 a b 



26c/- af 2 -ce 2 = (b 2 - acY— = (b 2 - ac) %-, 

 c b 



be-af=(b 2 -ac)f =(6 2 -ac)i. 

 c b 



When both conditions are fulfilled, the values of £ and r\ are more general 

 than when one only holds. 



