Rev. G. Phillips ofi the Summation of Series. 437 



And 



^yJy Jy-yJy y Jy-^-y v^.i.2,...n 



yn + l n+2 



+ i;r^^ , + ^ — — , + &c. (1) 



2'". 3.4...?r+T 3'«.4.5...« + 2 



Let / — / — ... ?n—V/ = iv^ , 



^yy^yy Jy-^-y """' 



Suppose then 4^_2=/^«^_3 



'''\-lfy'^ra-X= y-^-^fy.'Wn.-i "fy. y\-Afy'^ra-, 



•^ n-^Jy m-\ 2 n-ijy'^m-l ^2.3^-5/y.m-l 

 -&C. 



"^ 1.2.3...«-1^' ^Jl/.Wm-l 



+ ^^^^..^ Cy'^-'^'W , (2) 



- 1.2.3...«-2^i/'^ "'-* ^ ^ 



Now if this be taken between the limits of j/ = and v = 1, 

 we have 



1 1 1 



l"'.2.3...w "^ 2'".3.4...^+T "^ 3"'.4.5.6...^+fl''' ^^' 



_ 1 1 1 



" 1-2.3...^^'^ 2-.3.4...« "^ 3-4.5. e-.T^Tl* ^ ^^' 



2 ll'»,2.3...n^^'^2"'.3.4..In^^"''3"'.4.5...« "•" ^^'J 



1 / 1 1 1 



+ 2.3Vi'».2.3...«^"^2'".3.4..;:«^"^3'".4.5...'«^^"^^^*) 

 - &c. 



- 2.3...^^^^1'"-2.3 ^ 2"'.3.4 "^ 3'».4.5 "*" / 



■^ 2.3.4..r»^^ I'^.S ''" 2"'.3 "^ 3'».4 + °'*^'/ 



Also 



