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DlMONSTRATION» 



Let X, x + x, X + 2X - - , * + ^ |be cotcmporaneour 

 z, z', z", z\z' - z + x\ v^l"^^ °^ '^ ^""^ "" 



Let alfo a, a', a, a &c. ^ , r 



, ,, ,„ „ be difFerences of the refpec-- 



i>, b, t> , &c. J 



d, &c. j 



Then by the theorem for differences. 



n. n — I , n. n — i. n — 2 , « ^^^^ 



z ■\- z = z -\- na A b + c +, &c. where- , 



I. 2. ^ I. 2. 3 



n is the number of fucceffive values from ^ to x' + x, or from z to ^ 

 z ^z. Now if n be increafed Jtne Htntte, any afligned number of 

 terms of this quantity approaches to the fame number of terms in 

 the feries, 



n"" b , n^ c ^ ^ . ,. - , r ^4.^ 



z 4.na + T H, &c. as its bmit, or becaule « = — , to 



^ I. 2. 1. 2 3- X 



/2 ' h V ^ 



its equal z~ + —x H X 1 1- &c. But when n is fo increafed, 



^ ■ X x' 1, 2. 



the limiting ratio of tf : x or the limiting ratio of the increments. 



of< 



