122 Prof. Young*s Investigation of FormulcB for 



to twice the series whose general term is B^, minus the leading 

 term in the other series ; both series being, of course, regarded 

 as commencing at the same value of n. If this value be unit, 

 then, calling the series whose general term is B^, S', and that 

 whose general term is A*, S, we shall evidently have 



S --1-/28' ^ \ 



2 

 minus — g—j, the series whose general term is 



1 



(D.) 



} 



w (w + 'pf (ti + 2pf.,. [w + (w — l)pY(n + mp) 

 But it is evident, from the original relation 



A = — (B - C), 

 that the fraction D is equal to 



_L/ I 



mp\n{n-{- pY {n + 2;?)^.. [?i + (m — l);;]^ 



1 



{n + pY {n + 2pf.,» [// + (m — 1) pY (n + mp) 



so that the entire series, whose several terms are generally re- 

 presented by D, will be equal to , the difference between 



two series whose corresponding terms are generally expressed 

 by the two fractions within the brackets. If, however, we re- 

 gard the latter of these series, that is, the subtractive series, to 

 originate at the immediately preceding term, instead of at the 

 term where it actually begins, and then perform the subtrac- 

 tion, we must add to the result the leading term previously in- 

 troduced ; that is, supposing the leading term in each series to 

 have w = 1, we must add the fraction 



12 (1 + pY (1 + 2pY ... LI + {m- l)pr 



Now the result of the subtraction spoken of is readily seen to 

 be — (w — l)pS' ; and consequently, by introducing the above 

 correction, the true difference between the two series will be 

 expressed by 



- \(rn-l)p S- 12(1 +p)m + 2_p)^.. [1 +(m-l)p]}- 

 It follows, therefore, that the value of the series S must be 



