Tables for the Solution of cm Eqiiatiwi. 207 



multiplications by \x\r + 1 have been conducted in two different forms 

 to avoid the possibility of mistakes. Thus, for instance, in working 

 out I (5-2) and 1^5 -2), the factor Jx/8, or 2 -6/8, can be used as it 

 stands, or as J, and also put into the form -| + i. The adoption in all 

 cases of two quite different processes is an almost infallible guide to the 

 detection of a mistake. 



7. The values of l Q (x) and I^x) being thus obtained, K (x) and K^z) 

 can be derived. 



We have 



o = 



Adding 



K (;r) = -I (* 



{/3 4 (S 2 - 1) + /3 6 (S 3 - 



It will be convenient to denote /?2r(S r -l) by ^ ne symbol y>2 r - 

 Hence 



KO(*)= -I W{log-E-l}-l + { 74 + r6 + 7s + ...... } ......... (10) 



The value of I (#) is known and that of log x can be found from 

 Wolfram's Table. The quantities y 2r must be derived, each from the 

 corresponding /3 2 r- 



For earlier values of y 2r the multiplier S r - 1 is most easily used in 

 the natural form 



111 1 



2 + 3 + 4+ ...... + F 



For later values, it is simpler to use the decimalized values of S r - 1 

 given in Table V. 



In using the primary form, many simplifications are possible, thus 

 + -| =| = ^-f, and the multiplication is effected by shifting the decimal 

 point one place to the right, and dividing by 12. 



Again, + + = 1+^, 



and so on. In the computation of the values given in Table I, two 



