346 VAN orstrand: exponential function tables 



The first table by Newman contains 370 18-place values of 

 the descending exponential between the limits 0.1 and 37.0. This 

 is hardly the equivalent of a 16-place table as the original com- 

 putation included only 18 decimals. His second table contains 

 12-place values of e-' at intervals of 0.001 from x = 0.001 to 

 X = 15.349, and 14-place values of the same function from x = 

 15.350 to a; = 27.635, the interval being 0.002 from x = 15.350 

 to a: = 17.298, and 0.005 from x = 17.300 to x = 27.635. The 

 formula used was 



M±A^ = e-^"' = e-^[l±/i4-^=^^+- • •], 



2: 3: 



wherein h assumes the constant values 1, 0.1, 0.01, .... 

 dependent upon the interval of interpolation. Having given 

 e-^ and e^^ + "> the value of e-' - ^ is computed from the formula 

 by putting 



— e-' and iV = > - -' 

 m: ^^ n: 



ikf = y — e-^ and iV = > - e~\ 



m being an even and n an odd integer. The values of the 

 separate terms in these expressions may be computed by suc- 

 cessive divisions. Then the appropriate summations give 



M ^N = e-' + \ 

 a known quantity, and 



M - N = e~'- ^ 

 the quantity to be determined. The equation for M + N, pro- 

 vides a check on the values of M and A^, but the difference which 

 is the quantity sought is not verified by this method until an- 

 other interpolation is made. 



Glaisher gives 10-place logarithmic values and 9 significant 

 figures of the natural values of both the ascending and descend- 

 ing function for the following ranges of argument: 

 From X = 0.001 to a; = 0.100 at intervals of 0.001 

 '' X = 0.01 " X = 2.00 " " " 0.01 



" X = 0.1 " X = 10.0 " " " 0.1 



X 



= 1 '' a: = 500 " " " unity. 



Since the natural values were computed from the logarithmic 



