7-PAKTITION8 OF X. 149 



This proves that the value of 7 P* is always the integer 

 nearest, above or below, to the sum of the terms in x, c, y ; 

 af, andyj_i, &c., being neglected; and I suspect, from the 

 look of the expression, that c and y may be neglected also. 



The terms free from x in 7 P X can be reduced to a cir- 

 culating constant of 420 terms. The shape in which I leave 

 it has at least the advantage of occupying less room, and of 

 showing more of the structure of the function. I am content 

 to lay before the reader a definite expression of these 7-par- 

 titions of x, and am convinced that 8 P* can be deduced from 

 it with ease by the method here given ; and I think this can 

 be done without the labour of writing out at length this 

 tedious constant in 7 P X . 



The portions (gP*) and (gP*)' of 8 P* are very easily ob- 

 tained from the equations 



( 8 P z y-( e P^)'=[ 7 P^]'4-[ 7 P^ 9 ]'+[ 7 P_ 16 ]', 



by a process exactly like that employed to find ( 7 P X ) and 

 (7P*)'. The sum ( 8 P*)" of the circulating constants in 

 7 P x _i-r- 7 P x _9 4- ... is to be found from 



(8P,)"-»(sP^o)"=[7P^i]"H-[rP^]"+ • . • +[A-»]\ 

 the right side of which is the sum of 105 out of the 420 

 elements of the constant in 7 P, . Hence the sum ( 8 P*)" of all 

 these will be of the form 



and can be found easily before the constant in 7 F X is fully 

 written out. But I hope that a still more expeditious mode of 

 determining S will be discovered. 



In like manner the 9-partitions, 10-partitions, &c. of x may 

 be deduced, each almost by simple transcription from the last 

 preceding, if the constant of this be written out from the 

 formula arising in the process. 



