Tables for the Solution of an Equation. 



209 



2/ 



Hence 



The calculation therefore only involves two operations which entail 

 much labour, namely, the multiplication of I^JB) by {logo: -E - 1 

 and the computation of the series. 



9. The quantities given in Table I have been computed by these 

 formulae. The multiplications of I (#) and I^x) by (log- E - 1) 

 have been worked, taking (logic - E - 1) as multiplicand, which saves 

 a good deal of labour, as many of the lines of multiplication used in 

 finding K (#) occur again in finding K^x). In all cases the multiplications 

 have been carried to several places further than are used, or given, in 

 the final results. 



10. A process of verification has been applied to the values given in 

 Table I, based on the following theorem. 



It is easy to show that if y y : and y y. 2 be any two different 

 solutions of the fundamental equation (1), 



dx 



