208 Mr. W. Steadman Aldis. 



different processes for computing each y have been employed, so that 

 any mistake -Is almost certain to have been detected. 



For the lower values of x a second process of calculating the values 

 of j^r has been found from the obvious fact that if y' 2r be the value of 

 yzr when x becomes x/m t 



y. 2r = y- 2r . 



m -2r ' 



Thus the values of the quantities y for x = 2'6 can be deduced from 

 those for x = 5*2 by a series of divisions by 2 or powers of 2. 



For the values of x from x = 3'1 upwards, this process was not 

 available, but either two different transformations of the sum of the 

 vulgar fractions, or one such transformation, and the decimalized value 

 have been used in every case. 



Another process, which has been occasionally used, when the fraction 

 \x\r + 1 happened to be in low terms, is based on the easily proved 

 formula 



It only remains to multiply I () by (log x - E - 1) and, adding unity 

 to this product, to subtract the sum from 2y. The value of K (:), 

 which is always a positive quantity, is then obtained. 



8. The second function K^) can be readily expressed in terms of 

 quantities already found. 



For 



also 



_ I x f*tf L_ f^f - 1 1 



*\W - \ 2 + \2l 11(1)11(2) + (2) 11(1)11(2) ' ' J 



adding 



r 

 2 1 



n(r)n(r+i) 



+ 



but / ; y Oj -i- S 2 -- 2 l /*"V ^ ^ R 



'*' H(1)H(2) = 5 \2/ 11(1)11(2) == x ' ^ 4 



H(2)n(3) ~ \2/ ' n(2)H(3) 



