﻿364 Sir James Cockle on Fractional Criticoids. 



wherein X and fju have the respective values written opposite to 

 them in the group (1), and 



6 = a( P + Z*)+2b(cr + £ V ) + c(T + < n *)+2eZ+2f7 ] + Sr , 



or, recurring to the notation of art. 3, 



0=fl/) + 25<r+CT + <£(£*7). 



In the critical transformation both A and fju vanish. Hence, 



by (2), 



0=ap + 2bs + CT + ef;+fy+ff. ... (4) 

 Again, 



d\ _ h da ^ db de 

 dx " dx dx dx 



Consequently 



du. 7 dbf.dc, df 



f x + f y= 0=a P + 2 bs + cr 



(da db\ £ (db ,dc\ de df 

 + \a^ + du)^ + \dx + dyr + Ix^Jy 



dy/* \dx dy) dx dy 

 Hence, substituting in (4), 



>, / da db\ f. , / - db dc\ , de df ,,,. 



and this will be the criticoid, provided that the values of f and 

 77, given in art 3, satisfy the condition 



d%_ d*£ = d*£ = drj 



dy ~~ dy dx dx dy ~~ dx 



Since f is not, in general, obtained by the solution of a differ- 

 ential equation, it is necessary to test the desired transforma- 

 tion by the condition 



%=% •- < 6 > 



When this condition is satisfied we have, for determining f, the 

 equation 



If (6) is not fulfilled the transformation cannot be effected. 



7. In dealing thus far with the quotoid none of the conditions 

 (3) are supposed to hold. Now, assume that be—qf=0 alone 

 obtains. Then 



