Tables for the Solution of an Equation. 217 



A slight additional verification of the general accuracy of Table II 

 has been gained by the calculation of the term /3 for the values 8, 9, 

 10, 11, and 12 by elementary arithmetic and the exponential theorem 

 without the use of logarithms. 



The last figure of the quantities in Table II eannot be depended on 

 for strict accuracy, in which respect the table differs from Table I. 



20. A farther extension of the formulae of Articles 17 and 18 has 

 some interest. 



If, with the same notation extended, the quantity (u 5 - u_ 5 )/lQh be 

 denoted by e, it is not difficult to prove that 



14 126 



This value of u can most easily be computed by subtracting 



-3 



126 



from the value of u given in (18). 



This farther correction is too small to be applied with any certainty 

 to the values of K 1 (a;) derived from K (x) in Table II. Obviously 

 however, all these formulae may be equally well applied to Table I, and 

 throughout the range of that table, this formula deduces a value of 

 K^x) more accurate to one or two places than that given in (19). 



To give two examples ; one from the earlier part of the table. 



If x = 2-6 



Equation (18) gives - K^x) = 0'065 284 052 521 550 



(19) -K^X) = 0-065 284 044 927 362 



(21) -K^x) = 0-065 284 045 062 511 



while the correct value is 0-065 284 045 058 531 



Again taking the largest value of x in Table I which admits of the 

 application of (21), namely x = 5 '5, 



Equation (18) gives - K^x) = 0'002 325 569 051 888 

 (19) - K^a;) = 0'002 325 569 008 660 

 (21) ii - K i(*) = ' 002 325 569 08 85 



while the correct value is 0'002 325 569 008 849 005 



None of these formulae is sufficient for verification of the values in 

 Table I to the last figure given. 



