14 MR. E. CUNNINGHAM ON THE NOEMAL SERIES 



9. Consider the system of equations X 1 derived from X by changing 6J into 6 f > 

 and x into x 1 , viz. : 



X 1 . K+i-^VOxo 1 = 0, 



and let these be treated in exactly the same way as the equations X, the undetermined 

 elements of x 1 being supposed the same as those of x. 



From the two sets of equations X, X 1 let a new set be formed by subtracting 

 corresponding members of X and X 1 and dividing each remainder by P -0 P 2 , and let 



this new system be denoted by 



X-X 1 _ Q 



The expressions for x 1 obtained from X 1 in terms of 1 p + l ... will be identical with 

 those obtained from X in terms of the same with p l changed into P ". 



xx l 

 Let A P .T denote the expression ' ' 



Vp Up 



Then A ;) X is the system of equations 



= 0, (+, - 0Vi) A/d ~X = 0, 



a 1 /9* ^A / y-l-//v $ \ \ T:-. T,-, = 



\ w+l " p + 1 / *-*/>'^'2 ' \ p u p I *-*p'* J l * JU \ v ? 



Further A p (;r 1 ) = [ M = y 0> and l p+1 = 



Thus these equations are identical in form with the equations Y, except that 2 p - k 

 is replaced by 6 l p - k , &>0. Thus if from Y the y's be calculated as in I., p. 13, and 

 from A P X the quantities \x be similarly determined, the only difference between 

 Aa? r+1 and y r will be in the substitution of x... for t/i 1 ... and 6 l p - k for 2 p - k , A;>0, and 



Thus if we substitute the values of A p a; thus found in ApX^! the result will differ 

 from Y r only by the same substitutions. 



In a similar way, denoting by A P Y the difference equations 



Y-Y 1 



P *-0 P 3 



A P Y will differ from Z r _j only by the substitution of y?... for z?, tf v - v for 0^..., and 

 0*+l for 0J 3 , and so for the remainder of the first e l columns. 



