Tables for the Solution of an Equation. 211 



figures, two of them certainly correct, which he has calculated. The 

 differences at the lower end of the table then become regular up to 

 the twentieth order. 



This process has not been applied to the K^ic) column, because the 

 writer believes that, granted K () correct, the verification formula 

 above sufficiently proves the accuracy of K 1 (). The values of the 

 quantities in Table I are believed to be correct to the last figure given. 

 A dot after the last figure indicates that it has been increased by unity, 

 the first figure omitted being equal to or greater than 5. 



11. Table II has been computed by means of the formula (9). 



The remainder after s terms in the series involves the integral 



where 6 is some proper fraction. 



Now whatever n may be, after a time n s % becomes negative. 

 When s has reached such a value, inspection of (9) shows that the 

 terms in the series thereafter are alternately positive and negative, 

 inasmuch as a new negative factor is introduced in forming each suc- 

 cessive coefficient. It is also evident that, from and after that point 



^ 



in the series, the quantity ( 1 + ) 2 is numerically less than unity, 



V Say 



and the remainder required at any point to give the value of K n () is 

 numerically less than the next term in the series. 



Consequently, after the alternation of signs has begun, the sums of 

 s terms, (s+ 1) terms, (s+ 2) terms, &c., will be a series of quantities 

 alternately greater and less than the value of K n (). As long as the 

 terms of the series diminish, it is possible in this way to obtain a set 

 of quantities, continually approaching one another, between alternate 

 pairs of which K n (x) must lie. 



For the values n 0, n = 1, (9) gives 



12. In K (x) the alternation of signs begins with the first term. 

 Hence the sum of 1, 3, 5, ... terms is numerically greater than the 

 value of K (a;), while the sum of 2, 4, 6, ... terms is less. 



The r+lth term is derived from the rth by multiplying by 

 (2r - l) 2 /Srz. As long as this factor is less than unity, the r + 1th term 

 is less than the ?*th, and the terms continue to diminish. The r+ 1th 



