﻿Sir James Cockle on Fractional Criticoids. 361 



cients, whether variable or constant, in the latter are as different 

 in their nature from the differential symbols to which they are 

 attached as the coefficients in the former from their accompany- 

 ing variables. Suppose that <£ is of n dimensions, and that, in 

 the quotic, x, y, . . are respectively changed into x-\-%, y + rj, . . 

 and that f , rj, . . are so determined as to cause the coefficients of 

 % n ~ l , y n ~ x , • • , which must be distinguished from those of xy n ~ l , 

 yx n ~ l , . . , to vanish. Call this a critical transformation, and let 

 the other coefficients, excepting those of x n , y n , . .x n ~ l y, xy n ~ l , . . 

 which are unchanged, be called, after the conditions of such 

 evanescence are satisfied, critical functions. When all the cor- 

 responding critical functions of two quotics are equal, each to 

 each, then, if the quotics are of the same degree, one quotic is a 

 transformation of the other. If they are not of the same degree, 

 then a transformation will render them uniform. 



2. When there is only one variable, critical functions have, in 

 the differential calculus, analogues discussed in my paper " On 

 Criticoids " in the last March (1870) Number of this Journal*. 

 These criticoids are obtained in consequence of the analogy be- 

 tween the linear transformation of a quotic in one variable and 

 the factorial transformation of a quotoid in one dependent and 

 one independent variable. Quotics with more than one vari- 

 able correspond to linear partial quotoids with one dependent 

 variable and as many independent variables as there are variables 

 in the quotics, and whereof the orders are the same as the de- 

 grees of the quotics. In pursuing the analogy between the 

 linear transformation of quotics and the factorial transformation 

 of quotoids, I shall confine this paper to quotics of the second 

 degree in two variables, and to the corresponding quotoid of the 

 second order with one dependent and two independent variables. 



3. Let, then, 



<£(#, y) = ax* + 2bxy -f cy 2 + 2ex + 2fy -f g y 

 and we have 



<l>(x + Z,y + v) = ax' 2 + 2bxy + cy' 2 + 2\x + 2!iy + (l>{Z,v)> 

 where 



\=a% + b7) + e,l 



^bg+crj+fj ^ 



Hence 



and when, as in the critical case, X and //, both vanish, the last 

 three equations give on reduction, 



* Phil. Mag. S. 4. vol. xxxix. No. 260. pp. 201-211. 



