208 Sir James Cockle on Criticoids. 



spectively become 



-J'+«? + « 2 =B 2 (35) 



-$ + 2«?-3« ia2 + « 3 =B 3 ; (36) 



while if by means of (27) and (2) we deduce 



u ^ a ~-^' < 3 '> 



and then, by means of (3?) and (27), eliminate u% and w, from 

 (34), we obtain 



•fl2^-^-^ 1 ^ + «4+^ 2 -a?-^ : l j 2 = B 4 .(88) 



Hence, comparing (35) with (24), (36) with (25), and (38) with 

 (26) and (24), we see that 



B 2 =CL ........ (39) 



B 3 =A, . . • (40) 



B 4 =V+3D 2 ; (41) 



and it will be observed that, in virtue of (27), when the trans- 

 formation is such as to cause the second term of the transformed 

 quantoid to disappear, then B 2 , B 3 , . . B m are the successive co- 

 efficients of the transformed quantoid divided by their respective 

 coefficients of binomial development, viz. 



n — I Ti—1 n — 2 



9. Thirdly, these results may be accounted for, and an inde- 

 finite series of similar results anticipated, a priori. To this end 

 I recur to the critical functions of algebra, which may be regarded 

 either as functions of the differences of the roots of an equation, 

 or, independently of the existence of any root of any equation, 

 as results of transformation. Regarded in the latter point of 

 view, there is a mode of deriving a critical function from a quan- 

 tic which suggests an analogous mode of deriving a criticoid from 



