Statistical iiieclianics 75 



«l) = cosl|î-l + ^--^+..4 + 



+ sin 1 \B 



2 2 ^ 4 4 



where * 



(180) -Kl] 



— X 



e dx 

 1 + a;2' 



0 



The computation of this integral may be carried out with the lielp of mecha- 

 nical quadrature. 



Let now Si** a; and Ci a? be defined by the series 



X 



^ 313 ^ 515 717 ^ / } ' 



(181) 



then we have 



(182) ^^{x) = cos ^1^^ — Si x-j + sin x Ci x. 



The functions Si x and Ci a? are tabulated by Glaisher in Phil. Trans. London, 

 T. 160 (1870) (in part reproduced in Jahnke's »Functionentafeln»). It may 

 be observed that (in both places) the relation between the integralsine and the 

 integralcosine • and the integrallogarithm is incorrectly formulated (the relation of 

 Glaisher and Jahnke is not valid for large values of x). 



A series for i^{x) valid for large values of a; is obtained in the following 

 manner. In the differential equation 



(Py _ 1 



we put 



0 



and get the following recursion formulae for the coefficients: 



«2r+l + (2r 1) 2r 0!2r+l = 0, 



«1 = 1: 



* The resulting value of B is, according to Schlömilch 5 = + 0.5772156. 

 ** This function (first introduced by Schlömilch) differs somewhat from the function si x, 



efined above, the relation between them being si cc = Si cc — ^ . 



