K-PAKTITIONS OF N. 131 



aPx-3= 2Px-4+«Px-7+ . . . +2PC-I, C\ 



aP* — aPx-a^^aPx-i* ^' 



The function 3?^ can be found from (c) by summing SsP,** 

 which is by (E), 



or to prevent any ambiguity arising from the three values of c, 



between the proper limits. This labour, however, we can 

 easily avoid ; but it is useful to observe that we are to expect 

 in our results terms multiplied by *^x-ii *3^_2> and *3,, as 

 well as by *2^ and *2^^i. 



Every term iV^-/ of equation (c) on the right side has an 

 ordinary algebraic portion, viz. \{x—f). If we sum all these 

 in both (c) and (c'), and denote those sums by (aP,) and 

 (aPx-a), we shall, if we subtract the latter of these from the 

 former, have remaining the algebraic portion of 2Px-i> which 

 we may denote by [2PX-1I j that is, we obtain 



(,P,)-(3P,-,) = [.P,-,]=i(X-l). (f. 



As not more than one power of x can disappear by this 

 subtraction, and the first only is found in the remainder, we 

 know that 



(aPx) = Aa;^^. 3^^.^, 

 — (3P,-3)= — Aa:2-h6Arr— 9 A 

 -- B^+3B 



