30 ME. E. CUNNINGHAM ON THE NORMAL SERIES 



Similarly x l * = xt* = Q, and so on. 



The equations for the second column are 



I. (1) y 2 1 -i 1 -X = ) (2) 3-0 2 -Zi 2 = 0, 

 (2) gives 2 = 2 = 1 + 0i and (1) gives J/2 1 = A. 



II. (1) yj - (\ + O a ) yi l - z, 2 - A- xS = 0, (2) (2 - 0,) yj - xf - W = 0, 

 (2) gives A. = and (1) gives y? = 2yJ and also yj = 0. 



III. (1) y? - %! 2 - zi 3 - W = 0, (2) (1 - 6 S ) </2 2 - 2/1 1 = 0, 



so that yi 1 = and ?/ 2 2 = 0. 



It is easily seen that all the remaining terms vanish. 

 Thus the solution reducing at x = x to the matrix unity is 



1/1 o\\/i o\-i 



2J/U V 



where 



Thus 



fl 



f n r /o i\ 



T-lo *- tt o ^ bo oj 



16. The number of conditions found in the course of the analysis shows that the 

 solution in this form which may be called the " normal" form, by analogy with the 

 name "normal integrals" of linear equations is by no means always possible. As 

 many writers have pointed out, there is a much more general type of solution than 

 the normal series for the ordinary linear equation, in the form of a normal series in a 

 new independent variable x 1 '*, k being a positive integer (CAYLEY, HAMBUBGER, 

 FABKY, &c.). 



The method developed in the foregoing is peculiarly adaptable to the investigation 

 of these integrals, inasmuch as the transformation to a new independent variable is 

 very simply efiected. 



