Tables for the Solution of an Equation. 



With the help of these and the logarithm of x, that of /3 can be easily 

 ascertained, and then, if the logarithms of in 19 m 2 , m 3 , ...... , be tabu- 



lated, it is easy to derive those of fi v /3 2 , j8 3 , ...,.., in succession. 



The logarithms of m v m 2 , ...... , as far as it has been necessary to 



use them in the construction of Table II, are given at the end of this 

 paper in Table VI. 



14. In going through the calculation, it is, of course, useless to take 

 the values of the quantities^, /3 2 , ...... , to a decimal place further 



than the last one which can be accurately obtained in /5 . If ten-figure 

 logarithms be used, ten significant figures can be ordinarily obtained 

 with accuracy from the logarithm. Of this the writer has satisfied 

 himself by working out the value of (7r/2x) 1 '~ x by elementary arith- 

 metic and the exponential theorem, for one or two simple values of x, as 

 x = S, x = 11, and comparing the result so obtained with that derived 

 from the logarithms. They always agree for ten places, sometimes for 

 eleven, if account be taken of the second differences of the logarithms. 



It follows that for larger values of x, for which the smallest term in 

 the series is less than 10~ 10 /3 , the value of K (a?) can be obtained with 

 accuracy, probably for ten, and pretty certainly for nine significant 

 figures. The tenth figure may be in error owing to the accumulation, 

 in addition, of the errors in the last places of the quantities /3 19 /2 2 , ...... . 



15. Equation (13) gives 



3 3-5 



The multipliers, disregarding sign, by which the coefficients of the 

 successive powers of x within the bracket are derived, each from the 

 preceding, are 



3 5 21 45 

 8" TB" 2~4' :j 2' 



Let these be denoted by the symbols fi v //, 2 , /x g , ...... , and let a series 



of quantities fi' v /3' 2 , /3' 8 , ...... , be obtained from /3 by the successive 



relations 



'i * /h/V- 1 . ft = /^ftor 1 , ft ~ 



having the same value as in Article (13). 

 Then evidently 



-[(A, + /3'i + /3's + ...... )-(/3' 2 + /?4 + ...... )]J 



the summations being carried on, either until the smallest term of the 

 series is reached, in the case of the lower values of x, or until a term is 

 arrived at which is less than 10~ 10 ^ , which will happen first for larger 

 values of x. 



