140 EEV. T. P. KIRKMAN ON THE 



It is to be observed that the largest of these is <4=905, a 

 number less than J*2880 ; whence it appears that 5?* is always 

 the nearest integer to the sum of its terms in x. 



We may now briefly deduce gP-r, which is made up of (gP^) 

 a sum of algebraic terms, of (gPx)' a sum of terms containing 

 circulating co-efficients of x^ and a sum of terms of the form 

 c?'0.*6O^_0. Writing first 



= (gP,)_(gP,.6) 



""I— A(a?— 6)5— B(^— 6)4— C(a;— 6)3-D(a;— 6)2— E(a;— 6) 



— F: 

 we obtain the equations 



^'^2880' •'•^""30*2880 

 B-6-4— A-10-62=A_. ...B=^ ^ 



'2880' 4 2880 



C-6-3-B-63+A-10-6«=~^; -.-Cr:^^ ^ 



2880' * ~ 9 2880 

 D-6-2_C-6=-3+B-63-4_A-5-6*==|?; .-.0=0 



E.6_D-6^+C-63_B-6HA.6-Ji; .•.E=-f.^„ 

 Wherefore 



V6^«^;-2880(20^ 4 T 9 — 30 f+^- 



Now since, x being = ^e'\-c^ and c^7, gPar-e ^^^ "^^ terms 

 at all, we must have 



(gP,)=[5P,_i], or 



1 J^'4.3^V^-.^^'Uf-— Irc-iy+iorc-iv^ 



2880130+4 + 9 30 |^^ -288o|^'' ^^ i-lU(.c-i; 



-|-10(c— 1)2— 30(c— 1)1. 



