302 COMPUTATION OF THE VALUES OF LOG G'" 



Hence, we have 



log^ (1 + O = *>' (\x) + f ftAO 2 -\ i AW} 



^ \Zi/ i. A I A t i 







+ &c. 

 observing that A"0 m = when n is greater than m. 



The above may be symbolically expressed thus 



or 



where the symbols of operation A, A 2 , &c. act only on the quantities 

 denoted by O 1 , O 2 , O 3 , &c. 



This again may be symbolically represented thus 



log J(l + **) = log (l + J A) [e 01 * - 1], or by log (l + \ A) e' z simply, 

 ^ \ ^ / \ * / 



since A, A 2 , &c. performed on the constant 1 produce zero. 



If we substitute the known values of A"0 m as given in Herschel's 

 "Examples of Finite Differences," p. 9, we have 



