32 ME. E. CUNNINGHAM ON THE NOEMAL SERIES 



that E! = n. It will be shown that a subnormal form satisfying the equation certainly 

 exists if a/" i= with k = n. 



We know from the foregoing investigation that 6\ p is given as the root of an 

 equation of degree n, and that, if the roots of this equation are all different, no more 

 conditions than those just mentioned as satisfied are necessary to ensure the existence 

 of the subnormal solution. But in this case the equation for 0\ p is particularly easy 

 to construct. We have, in fact, 



nxf-0 1 ,, = 0, x? = xf = ... = a," = 0, 



nxf-ffi^' = 0, ir a = ... = x = 0, 



nxS-ffinStf-ffiv-M* = 0, xf = ... = x s = 0, 



r n Q\ n-l_f)2 -l _ A 



X B _i U ,, p Ji n -2 " np&n-X ... - U, 



u ,,, i x n n - l .. . + A n ( p _])+i = 0, 



from which at once we have 



-+I - u ' 



Tlius, unless A ln B(p _ 1)+1 = 0, the values of 6 l n) , are all different, and a solution in 

 subnormal form is therefore possible, as stated above, with the independent variable 

 changed to x i;n . 



If, however, A 1 ",,^-]^] = 0, we have 



(n _ _ tin _ f\ 



\J n p ... -- I/ n p - U, 



and it will be found that the same conditions are necessary between the constants 

 A.n(p 1)+1 as were found previously (p. 13) between the constants ot p , A n(p _, )+1 

 being the same as n . a. p , e.g., from the equations 



- 1 .^,,*, = 0, A. 2 '\ (p - l)+l -nx n = 0, 

 we have 



1 Mp _ 1)+1 = 0, i.e, a p 2 ' n + a p 1 '"- 1 = 0, and so on. 



Consider now what happens when these conditions are not all satisfied. Suppose, 

 for instance, a/^ + a/'"- 1 ^= 0. Let the original system be transformed by the change 

 of variables 



Then from the equations 



= z k , k = nl. 



, (? ,- 1)+1 2, 2 = 0, 

 we obtain the equation for 6 l kp 



