[MILLER & rosebrugh] VALUES OF FUNCTIONS INVOLVING e-x 77 



which may be deduced by Taylor's theorem, treating intervals of 0.001 

 as units.' 



The computation was carried out with ten places of decimals, and 

 every tenth value was checked by Glaisher's results, and found to agree 

 within a few units in the last place ; the tenth place was then rejected. 

 Incidentally the agreement furnishes evidence of the accuracy of Glaisher's 

 Table, at least as far as the ninth decimal place, and of our Table of e-^/x 

 as far as the seventh. 



Following Glaisher,^ the integration constant 



;/ = 0.577 215 665 

 was adopted, whence Ei (- 00 ) = 0. 



Values of (^ dx -\- loge x from x = Q to x = ^ . 100 at intervals of 



X 



00 



.001 {Table I). As for x = 0, /^ (ijj becomes infinite, interpolation for 



X 



low values of x would be impossible. By adding locje x however, the 



00 

 infinity is removed and this difficulty is avoided, /.fll dx -\- log^ x for 



00 

 a; = having the value — ;/. By subtracting the logarithm, f^ dx 



X 



may be obtained. 



The values in the Table were computed in the same manner as those 

 00 

 0Ï I l^dx, U8ing^_fzi3 in place of_^, and every tenth was checked against 



X 



Glaisher's value for P^ dx. The numbers so obtained are negative. 



00 *^ 



For example : 



00 



for X = 0-050, /!£! dx + log^ x= — 0.527 833 785 

 «^ X 



X 



CD 



hence J^dx = — 0.527 833 785 — log^U.Qhi) 



^ ^- = — 0.527 833 785 + 2.995 732 274 



= 2.467 898 489 

 00 

 Values of / £^dx from x = 0.100 to x = 1. 000 at intervals of .001 



1 The coefficients are those of the powers of x in the expansion of -j .. ^ 



See : Boole, Finite Differences. 



2 In Glaisher's Table the limits are the inverse of ours, hence the minus sign 

 prefixed to his numbers. 



