ON CALCULATION OF MATHEMATICAL TABLES. 245 



The Exponential Integral, Ei ( ± x). 



— X 



f ~ u 

 Several tables* of the function Ei (x)= - — du for both positive and negative 



' J m 



CO 



values of x, and of the Sine and Cosine Integrals, Si (x) = du and Ci (x) 



f co cos u 

 =— I — • du, have been published. 



Dr. Glaisher's Tables give 29 values of the functions to 18 places for x= 0-00 to 

 bOO, to 11 places for .r=l-0 to 5-0 by intervals of 0-1, and for integer values from 

 5 to 15 ; the Sine and Cosine Integrals are also tabulated to 7 places for a number of 

 integer values of x beyond this range. 



The series in descending powers of the argument, 



/, 1 ! 2' 3 ' 4' \ 



\ X X 2 X* x* 1 



is most suitable and convenient for calculating Ei (a:) for large values of x. When 

 x is negative, the signs of the terms are alternately positive and negative ; in this 

 case several places of decimals can be added to the result obtained by stopping the 

 calculation at the least term. If x=n-\-h, the divergent part of series can be repre- 

 sented by the product of the least term T and the factor cp;, of which the first five 

 terms are 



s-A(H^a + t+*'Ki(K-») 



For example, when £=11, (p t = 0-488898908 . . . and the least term of the series, 

 T=0-000139905948, the product, 0-000068399865, is equivalent to the divergent terms. 

 The result is, in this case, improved to the extent of eight or nine places of decimals. 

 If x is positive and equal to w+oc, the " converging factor " is 



For a=0, i.e. when * is a positive integer, the second least term must be multiplied 



by 



_4 4 + 8 16 

 3 135w 2835n' J 8505w 3 



When x=10, Ei (10) is given to four or five significant figures when the calculation is 

 restricted to the convergent terms of the asymptotic series ; by the above method 

 Ei ( 10) has been computed to twenty places of decimals, giving a result in agreement 

 with the value of the function in Bauschmger's Tables. 



*Bauschinger. Archiv der Math. u. Phys., 1843, pp. 28-34. 

 Bretschneider. Zeit. fur Math. u. Phys., vol. 6, 1861, pp. 127-139. 

 J. W. L. Glaisher. Phil. Trans., vol. 160, 1870, pp. 367-387. 

 J. P. Gram. AJhandlinger der Kopenhauencr Akad. (6), vol. 2, 1884, pp. 183-308. 

 Lord Rayleigh. Proc. Roy. Soc, A, vol. 90, 1914, pp. 318-323, or Collected 

 Papers, vol. 0, p. 228. 



