334 NEW SERIES for the 



we put I (*+ = x > !t follows ' that 0T0 = trv 



In like manner, putting I ( x T + — 7 J = X', and I ( x~* ) 



- # 2 x 4 



v-+v~ x ; +i ' 



= X", we have ( 1, 1 1 = *£l2, and | -1_! j =|£=?, 



&c. Again, becaufe X'= I ( x *+—,), therefore 2X' = 



2 >• x T ' 



- f * + -) + i, but -^x + i^=X, therefore 2 X' a = X-fi 

 and X'=v — — • In like manner, it will appear, that 



X 



" = n/ X + I , &c. 



2 



63. From the preceding inveftigation it appears, upon the 

 whole, that our fecond feries for the calculation of a logarithm 

 may be exprefled as follows. 



Putting x for any number, let a feries of quantities X, X', 

 X", X'", &c. be found fuch that 



x = t {»+ H , x<=^*±J, r^ &c. 



Then will 



1 

 log 2 # 



