] 42 "I KEY. T. P. KIEKMAN ON THE 



In this subtraction the constant G x must vanish of course, 

 if it be an ordinary constant ; and if it be a circulator, it can 

 depend only on the 7 values of c in equation (A) ; that is, it 

 must be of the form 



A # 7^+Gt- # 7 # ^+ . . . +G0-*?,, 

 so that Gjp=G x _ 7 , and can have only seven values. 



Equating the co-efficients of powers of x, we have 



42 A !_ . a- l • 



-157 2 A+35B= \l~ ,.-.B= ^ 



280 80 



20-7 3 A-10-7 ! B+4-7C= j^, .;. C= lw ; 



-15-7«A+10-7 3 B-6-7 J C+3-7D= - 1 -||2 S , .-. D= ~~ 5 ; 



H-7'A-5-7'B+4-7'CU3-7 2 D+27E=— ^g, .-. E= - ^gpj 



_7. A+7 »B-7'C+7»D-rE + 7F=^, .-. F= - ggj. 

 Wherefore 



^ 7 x) 7-720 2 " 1 "720 2 " t '720 2 " t " 7-720 2 2-720 2 14'720 2+ *' 



The value of the constant G x is obtained from the equation, 

 (putting #=7(l— l)+c) 



(7P C )=( 7 Pc- 7 )+[ & Pc- 1 ]=[ tt P,- 1 ], 



for 7 P X _7 has no terms at all on the right of equation (A'), if 

 #Z8 ; i.e., ( 7 P^ 7 )=0. This constant therefore will be a cir- 

 culator of the form SG/.?^. 



The value of G 7 is readily found by the equation 

 ( 7 Po)=(7Po-7)+[eP^i]=0=G 7 ; 

 for #=0, which requires e=0, c=7, makes every term on the 

 right of equation (A) vanish. The remaining six values, or 

 all the seven, are obtained by putting for c its successive 

 values in 



( 7 P,)=[ 6 Po-i], 



