1\102 Mr, Adams on [Feb. 



By substituting for A, B, C, &c. we have 



^ - ^ -*- 1 + 1.2 +1.2.8 + 1.2.3.4 + ^^• 



Example 8. — Expand into a series the log. /^^^^-^) log. of base 

 unity. 

 u = /(^) = /. 1 +/a;= 0+/a: = A+/a:.-. A = 0, 



— = -^=- + /'r-B + /'x • B=- 



Tr> = (^ = 0+/n^=2C+/..a:...C = 0, 



_2 



^tt 48rta:.96«jr3 ^ ^ ooj-r«.^ 



7]^ = (^jiZl^a + (^e-:^^ = + /.v X = 2 . 3 . 4 E +/,vx .-. 



E = 0, V- ^•-- 



JT^ = (^zpys +A^ = IT +/vi X = 2 . 3 . 4 . 5 F 4-/v, :r .-. 



F-^ 



&C , 



By substituting for A, B, C, &c. we have ^ 



By adding the series in Example 3 to the series in Example 6». 

 or subtracting the series in Example 4, from the series in Ex- 

 ample 3, we have the same result as above, which is evident 

 from the nature of logarithms. 



Example 9. — It is required to expand into a series the expres- 



• sion . 



a — X 

 a™— X 



ti = = «*~* 4- f^ = A + /'x .*. A = «"•"', 



a — X ^ ' 



77 = ^. + /■ ^ = «""' +i; r = B + /, X .-. B = a-', 



l^.= l^,F ^^^' = 2 «- +y> = 2C +/.X.-.C = 

 a-', 



4l> U 2 . 3 (<!"• ~ X"») , ^ J, ^ ^_, , ^ O O T-k 1 ^ 



715 = — r-TTT— +/5 ^ = 2 . 3 a- ^ +/6 :r =r 2 . 3 D +/^ 

 X .-. D = a"-*. 



&c , 



By substituting for A, B, C, we have 



f!Ll£L = a-- • 4 dr~''x + a^-'x* H- a"-*j:' + &c. Or, 



