relating to Geometrical Series. 197 



But r"''- 1 = (r "••% 1 ) (/"'''+?) (r" + 1 ) ('■"' - 1 ) to 



7t + I terms ; 



r'^— 1 = (r f 1) (r^ + l) (r+l)(r-l) to w -f- 1 terms ; 



therefore, by substitution, in equation («), we have 



A ^ (r^V 1) (r^-'^'^'f 1) (r^-4-1) (r^-l) (r-l) 



(r-7l) (r^Tl) (r+1) (r-1) (r"-l)- 



By elimination and separation of the factors, 



n— 1 n—2 



r -i- 1 r +1 r + A y -f 1 



r^^-' + l r^'^-Vl 

 to w terms. 



Bui r+i =, ,.('»-l)2_,.(— 2)2 ^ ^2.2 _ ^.2^ J 



TO — 1 



r« + 1 



n-2 



&c. &c. &c. &c. 



r 4-1 m—l 7/1—2 



=r — r + r^—r-\-l,Gdch 



series to w terms. 



Hence by substitution in equation (Z>.) we have 



^+'-' +'-^"+ ' '=(l-r+r^ +r"'->-l 



l+r+r'+ r"*-' L / 



x{l-rV»" +,('»-»n 



l-/+^2V r("-'^' J- 



to w factors of m terms each. 



Or, as it may be more compendiously written 



n --n 



o^ 2.2' (f»-l)2" 2 



1+r +r +...+r 



l+r + r^+ +r'' 



-=|l~r+r^ r-»j. 



