132 KEY. T. P. KIRKMAN ON THE 



When X — 3^0, sP^-a in equation (c') has no terms what- 

 ever, and [cT] becomes, since x=0'S+c, 



(3P,)=[,PJ 



"12^ +^ 2"' 



12 12 ' 



for it vanishes for c=2, or c=3, and when c=l, it is — tV. 

 The portion (aPa?) of s^x is thus proved to be 



(3P.)- ^^ 



To find the rest of aP^, which for anything yet shown, may 

 also contain an ordinary algebraic portion, we have to sum 

 the (c+1) terms 



-ir2.-2+*2,.5+ • • • +*2,.i+*2,_2) 



If e+l=2A, or x=6h+c—3, 

 this is 



^*9 ^#9 __ ^_. 37+3— c 

 ~2' "^ ~2 "^^-'-""2 12"' 



for *2c_2 =1 for a? odd, c being then =2, and *2^^i even, c 

 being either = 1 or 3 : and the equation just written is true, 

 whether x be odd or even, as *2^+*2^_i=l, whatever be the 

 integer x. 



If €=2h\ or x=eh'^c, 

 the above sum is 



-i^'*2,-iA'*2,.i-i*2,_, 

 _ x—c 6 ^^ 

 12" "12 '^'^'' 



The sought remainder of gP-p is therefore 



— *2,_i-^(a;+3— c)— *2e-TV(^— c+6.*2c-2), 



Consequently, by addition of this to (sPx), (a;=3e+c), (c70), 

 3P^=^{a.2_*3^_r2~*2e-r(3_c)+*2,-(c_6-*2c-2)}- 



