﻿Sir James Cockle on Fractional Criticoids. 367 



Hence, when (9) holds, 



and the quotoid of art. 8 may be written in either of the forms, 



\A + 2eD + ag\z J 

 or 



But this last result admits of the symbolical decomposition 



(D+U)(D-H0*, 



provided that U and u satisfy the equations 



XT _ da b da 



U +u=2e — j -j-> 



dx a dy 



J)u + Vu — ag. 



Substituting in the latter the value of U obtained from the 

 former, transposing, and replacing D by its equivalent, we have, 

 for determining u } the partial equation of the first order, 



du T du 9 /_ da b da\ , 



a- r +b- r =u?—(2e—-j -j-xu + ag. 



dx ay \ ax a ay/ * 



11. Next, let A=0 be any solution whatever of bdx—ady=0. 

 Then, as we see by (8), the form of f will, in general, be f =/3 + B, 

 wherein /6 is a definite function of y and x, and B is an arbitrary 

 function of A. Moreover DA = 0, and therefore in general 

 DB=0 and (D + u)Bz = B(D + u)z. Consequently the trans- 

 formed and divided quotoid becomes successively 



€ -^-B( D _ { _xJ)(D + w)€^ B z=e-^- B (D+U)e B (D + M)6^ 



and it is seen that it is only the definite portion /3 of f that acts 

 effectively in the transformation. It may be noticed in passing, 

 that since, if (T) + u)z=0, then also (D + u)Bz=l$(D + u)z = 0, 

 therefore the solution of the partial linear and homogeneous 

 differential equation of the first order may be written in the form 

 £B 4-^8 2 B 2 whenever it admits of the form /3 + /3 2 B 2 . When the 

 condition (9) is fulfilled, then, equating the quotoid to a function 

 of x and y, we have a soluble differential equation. When (9) 

 does not hold, arbitrary functions enter into the criticoid, and 

 give a wide range of transformations of the quotoid last discussed. 



12. Thus the general quotic, wherein the coefficients are con- 

 stant, and the quotoid wherein the coefficients are variable 



