NO. B.] TABLES OF EXPONENTIAL FUNCTION— VAN ORSTRAND. 7 



the quantity to be determined. The equation for M± N provides a check on the values of 

 M and N, but the sum or difference which is the quantity sought is not verified by this method 

 until another interpolation is made. 



Gram ' gives values of the ascending exponential to 24 places of decimals at intervals of 

 imity between the limits x = and a; = 20, inclusive; also values of the same function from 

 z = 5.00 to x = 20.00, inclusive, at intervals of 0.02, the number of tabular decimals ranging 

 from 4 to 15; and from a; = 0.1 to a; = 15.0, the values are given to one decimal only at inter- 

 vals of 0.1. Some of the values wore obtained by either repeated multiphcation or logarith- 

 mic computation, and the remainder were borrowed from Schulze, Bretschneider, and Opper- 

 mann. 



Glaisher ' gives 10-place logarithmic values and 9 significant figures of the natural values 

 of both the ascending and descending function for the following ranges of argiiment: 

 From x = 0.001 to a; = 0.100 at intervals of 0.001. 

 From x = 0.01 to x = 2.00 at intervals of 0.01. 

 From a; = 0.1 to a; = 10.0 at intervals of 0.1. 

 From X = 1 to a; = 500 at intervals of unity. 



Since the natural values were computed from the logarithmic values, the maximum 

 tabular error is one unit in the ninth significant figure with the exception of values of e-^ con- 

 tained in Newman's tables. The remaining values of Glaisher's tables were checked either by 

 differences or by duplicate computation. Glaisher gives also the reciprocals of the factorials 

 from 1 to 50, inclusive, to 28 significant figures, and verifies his values by forming the summa- 

 tions for e and e-K A further verification is obtained by evaluating e"'" to 32 decimal places 

 by means of the formula 



* - 10" 

 The quantity y X 10~™ is here an approximate value of e~^. The equation gives 



loge(y + 7i) = n loge 10 - X 

 a known quantity. Since log^y is also known, we may evaluate the expressions 



Wog^iy + h) -log^y] and y]loge{y + h) -logey]. 

 The expansion of the first by Taylor's series gives 



Ji. = yWogtiy + 1)- log,;/] + 2 y ~ 3 p '*' ' " ' ' 



from which an approximate value of h y-' may be computed by neglecting terms in h beginning 

 with the square. Finally the substitution of 



1 Tv' , 1 , ^' 

 7; y -; and — -^h-z 



2 " y' 3 y' 



in the preceding equation gives a corrected value of ^. 



Burgess^ gives 30-place values of e~^ for x = 0.5, 1, 2, ... 10; and 14 values of «~*' at 

 irregular intervals between the limits 1.0 and 3.0, ranging in extent from 23 to 27 decimals. 

 These values were used in his evaluation of the probabihty integral, but no information seems 

 to have been given with regard to either method or accuracy of computation. 



In his "Rectification of the Circle" (1853), Shanks evaluates the Naperian base by direct 

 substitution in the series to 137 places of decimals. His second computation * was carried to 



' J. p. Gram, Undersogelscr angaaende Maengdcn at Prlmtal under en given Graense. Copenhagen Academy, 6 vol. II (1884), pp. 183-306. 

 « J. W. L. Glaisher. Tables of the exponential function. Trans. Camb. Phil. Soc., vol. 13 (1883), pp. 244-272. 



> James Burgess. On the deflnita Integral _?_ I e"**,!/, with extended tables of values. Trans. Roy. Soc. Edinburgh, vol. 39 (1900), pp. 

 257-321. -JrJo 



' William Shanks. On the extension of the value of the base of Napier's logarithms; of the Naperian logarithms of 2, 3, 5, and 10, and of the 

 Modulus of Brlgg's, or the Common system of logarithms; all to 205 places of decimals. Proc. Roy. Soc. Vol., VI (1850-1854), p. 397. 



