VAN orstrand: exponential function tables 347 



values, the maximum tabular error is one unit in the ninth sig- 

 nificant figure with the exception of values of e'"" contained in 

 Newman's tables. The remaining values of Glaisher's tables 

 were checked either by differences or by duplicate computation. 

 Mr. Shanks^ had previously obtained the value of e to 137 places 

 by direct substitution in the series. Glaisher verified this result 

 by means of the continued fraction^ 



e - 1 11 1 1 



2 l+6+10+---4n + 2H 



He also evaluated e-^" to 32 places of decimals, using the for- 

 mula 



e-=^+^ (1). 



The quantity y X 10"" is here an approximate value of e^"". 



Burgess gives 30-place values of e-* for x = 0.5, 1,2, . . . 

 10; and 14 values of e"""" at irregular intervals betwe'en the limits 

 1.0 and 3.0, ranging in extent from 23 to 27 decimals. These 

 values were used in his evaluation of the probability integral, 

 but no information seems to have been given with regard to 

 either method or accuracy of computation. 



The present table has been computed roughly to 35 decimals 

 or significant figures by means of the derivative formula 



A ten-groove computing machine of the millionaire type was 

 used in making the computations. In a few instances the eval- 

 uation was carried to 40 or more decimals for the purpose of 

 obtaining a complete check on my own values and the more 

 extended values given by Schulze and Glaisher. 



In order to obtain a check on the successive computations, 

 the value of the constant factors 6=*="-^ were evaluated from the 

 series with the aid of Glaisher's table of factorials. The value 

 of e is well known and its reciprocal to 44 places of decimals 



* One year later the value of e was given to 205 places. See Proc. Roy. Soc. 6: 

 397. 1854. 



6 Chrystals Algebra 2 : 495; A. Hall, Math. Monthly. 1861. 



