MEMOIRS NATIONAL ACADEMY OF SCIENCES. [Voc.xiv. 



0^ 



c- J sin a; J 1 z\ 1 i^ 1 x' , 



Six= / ^-dx=x-3 3-, + ^^-^^ + 



t/oO 



/t: I COS X , 



Cia;= / rfx 



Eix= / ^ dx=7 + |loge (x')+X+^|| + ^ jj + 



ri , 1 , 2! , 3! , 41, 1 



known, respectively, as the sine integral, the cosine integral and the exponential integral. The 



quantity y is the Eulerian constant 0.5772156 



Newman's ' contribution to the subject consists of the following: 



18-place values of e""^ from x = 0.0 to x = 37.0 at intervals of 0.1. 

 12-place values of e-^ from x = 0.000 to x = 15.349 at intervals of 0.001. 

 14-place values of e~^ from x = 15.350 to x= 17.298 at intervals of 0.002. 

 14-place values of e"^ from x= 17.300 to x = 27.635 at intervals of 0.005. 

 16-place values of e^ from x = 0.1 to x = 3.0 at intervals of 0.1. 

 12-place values of e^ from x = 0.001 to x = 2.000 at intervals of 0.001. 

 The 18-place table is hardly the equivalent of a 16-place table, as the original computation 

 included only 18 decimals. 



All of Newman's computations are based on formulas of the type 



M±N= e-*'i = e-[l ± ^, + J-^ ± |-' + . . .] 



wherein h assimies the constant values 1, 0.1, 0.01, . . . dependent upon the interval of inter- 

 polation. Having given e~^ and e-^+'' the value of e~^~^ is computed from the formula by 

 putting 



M= S^e-^ and iV= S-. e"^, 

 m! n\ ' 



m being an even and n an odd integer. The values of the separate terms in these expressions 

 may be computed by successive divisions. Then the appropriate sumimations give 



a known quantity, and 



' F. W. Newman. Tables o( the descending exponential function to 12 or 14 places of decimals. Trans. Camb. Phil. Soc, vol. XIII (1883), 

 pp. 146-241; table of the exponential function e' to 12 places of decimals. Trans. Camb. Phil. Soc., vol. XIV (1889), pp. 237-249. 



