146 REV. T. P. KIRKMAN ON THE 



+8424(( 7 -l)-(6 7 . 2 +6 7 _.)+( 7 ~15)(6 7 _ 16 H-6 7 ^) 



+(7-29X6^+6^)) 



-1176(( 7 -l)-6^+( 7 ~15)-6 7 _ 18 +(r-29)6 7 ^) 



+ 1674(( 7 -.8)-(6 7 _ 10 +6 7 _ 12 )+(7-22)(6 7 _ 24 +6 7 _ 26 ) 



+ ( r _36)(6 7 _ S8 +6 7 _ 4 o)) 



-7926 (•( 7 ~8)-6 7 _,+( 7 -22).6 7 _ 22 +( 7 — 36) 6 7 ^e) 



_7926(( r -l)-6 7 _ 1 +(7~15)-6 7 _ 16 +( 7 -29)6 7 _ 29 ) 

 +1674(( 7 -l)(6 7 _3+6 7 _ 5 )+( 7 ~15)(6 7 _ 17+ 6 7 _ 19 ) 



+ ( 7 ~29)(6 7 _ 31 +6 7 _ 33 )) 

 -1 1 76(( 7 -8)-6 7 _ n +( 7 -22)-6 7 _ 25 +(7— ^'Qy-J) 

 4-8424 (( 7 -8)(6y^+6 y - la ) +(7— 22X6^4-6^27) 



4-( 7 ~36)(6 7 _^ 7 4-6 7 - 41 ))J. 



In this expression it is to be borne in mind that 6#_ 7 =0, 

 whenever x/_l* 



We have now found, in the sum 



GP.)+(tP.)', 

 all the terms in x of 7 V X which arise from summing all the 

 powers 7O of x — 1, x — 8, &c., on the right of equation (A). 

 We have yet to determine ( 7 P#)", the sum of the constants 

 in the same member of (A). Of these there are 60 in every 



term 6 Pr-i-7a> of the form ^~ 2 X>f g *§Q x -g> of which, for a 



given value of x, only one has value, viz. one of the sixty 

 already given, p. 140. 

 We know that 



( 7 P,)"-(7P^2o)"=[eP^i]"+[ 6 P^]" + [ 6 P^ M ]"4- . . 4-[ 6 P_ 41 J". (D) 

 If we write z=420m4-60(n— I) + 04- 1 , we have, for 0=0, 

 x— 1=60^4-0, a— 8=60^4-53, x— 15=60^4-46, &c, so that 



the right member of this equation is —^ {y5 +./m +J& 4-J& 4- 

 ... to 60 terms J , that is, it is always the sum of the 60 num- 

 bers ,/J, the co-efficients of the circulating constant in 6 P*. 



