Co7nptitatio)i of the Functions GoO-O' ^^iC'^^)* ^^^^^^ J«(''\/'?^)- 36 

 Hence 



G,(X) = J,(:.){E+l-l0g;.;} + l-i^ 



The last portion of this = -i{ JoC'-) -I +(^2} = -^+1-'^, 



whence Gi(;i-) = Ji(.'t){E + 1 - loga}+?-| 



o;/! 1 1 1 1 Jo'» nr^\ 



5. The quantities ^ and y, and the multiples of the different 



values of (E + 1 - log 2') have been computed for the values of 



A', O'l, 0-2, ,6*0, in the process of calculation of Ko(r':) and Ki{x), 



given in the writer's former paper. It has been, therefore, an easy 

 matter to find by (7), (8), (9), and (10), the quantities Jo(x), Ji{x), Go{x), 

 and Gi{x) corresponding to the same values of x. The former two are of 

 course well known, but the recalculation affords a valuable verification 

 of the correctness of the quantities fS. The results are given in 

 Table I, appended to this paper, negative values being indicated by 

 the use of old numeral type. 



The formula used for verifying the values of I and K was 



l,{x) . K,{x)-l,{x) . Ki(<i)= I- 

 Replacing x by zi, by means of (4) and (6), this gives 



whence Ji(^) . Go(.-) - J,(z) . Gi(;^) = - ^ (H). 



This formula has been applied throughout Table I to each set of four 

 values, calculated to three places beyond those given. Where the 

 last figure has been increased by unity, in consequence of the first 

 omitted figure being equal to or greater than five, the fact is indicated 

 by a dot after the last figure. The column Gq{x) has also been tested 

 with satisfactory results by differencing. 



6. The value of In{x) can be readily expressed in terms of the quan- 

 tities (3, when n is either zero or unity, in one or two other cases, 

 beside those of a', being a pure imaginary or wholly reah 



