80 



Proceedings of the Roj/al Irish Academy. 



The coefficients of x^i/-, zx-y, %xy^ vanish since G is of the first 

 order in A ; F\.?> also of the first order, and C of the second order in X, 

 /(A.) being of the fourth order. 

 The coefficient of s^ is 



/'(A) ' 



••- K'C^-f{\;)^gG^^fF,. = gG,^fF,, ... /(X^) ^ 0, 

 .•. the coefficient of z* is zero. 

 The coefficient of 



but 



,,.y-^,[K^^^^^y 



F^G,. - C^, = 0, •.• f{K) = 0, 



The coefficient of 



z-x" = 5i* -J 



G, 



/'{K)o; 



B,C, - F,^ = 0, .-. -^ = ^, - - A^ + &c. ; 



and 



G^ 



•' 





■ 2^ ^^ 



xy z ^ + A^- 



Again consider 





(Fjjc + Gjii)z ^ n 



= 0. 



The coefficient of 





