76 ROYAL SOCIETY OF CANADA 



between .x = and a; = 2 (viz., x = 0.820, 1.28, 1.15, 1.93, 1.99) in 

 which there was a discrepancy of one unit in the ninth decimal place, and 

 the tenth digit was 4 or 5. 



There are two others in which the tenth figures are 2 and 8 respec- 

 tively, viz. : 



X Newman Glaisher 



1.55 0.212 247 973 827 0.212 247 973 



1.80 0.165 298 888 221 0.165 298 889 



In both cases Newman's result was found to be exactly the square of 

 his number for half the value of a:. 



Values of ~^, from a;=0.100 to a;= 1.000 at intervals of .001 (Table 



III), and from x = 1.00 to x = 2.00 at intervals of .01 (Table IV). 

 These were obtained by division from Newman's values of e-^. 



For use in calculating the Exponential Integral the first, second and 

 following differences of the first seven decimal places were taken until 

 they were small enough to be followed readily, these served also to detect 

 errors in the first six jDlaces. At the beginning of the table, x = 0.100 

 to ic = 0.210, the differencing was extended to the whole nine decimal 

 places. 



fi-X 



The last six figures of — were then multiplied by the corresponding 



value of X and the product compared with Newman's value of e-^, thus 

 eliminating errors in the last three places. This verification was per- 

 formed on the type- written sheets ready for the pri nter. 



This table was then used in computing / 1_ dx and / ^__ dx. 



X X 



Values of from x = to x = 0.100 at intervals of .001 



X -^ 



(Table III), By division, from Newman's values of e-^ ; checked by 

 taking the first and second ditterences on the type-written sheets. 

 These numbers were used in computing 



/» 00 -. 

 dx -\- loge *', and I ^ dx log^ x 

 X ^ X? ^ 



X X 



oo 



Values of fji^ dx from x = 0.100 to x = 1.000 at intervals of .001 

 •^ X 



X 



(Table I), and from x = 1. 00 to x = 2.00 at intervals of .01 (Table II). 

 It was found impossible to obtain these from Glaisher's table by interpola- 

 tion. The table was consequentl}^ built up from the values of e-^/x by 

 means of the relation 



^ . m (-X) = 0.001 (1 + i ^ - tV ^' + 2V ^' - t¥o ^')e-V^ 



