Statistical meclianics 78 

 Returning now to the function we have 



1 I ax e ax e 



2 i J \ X — i X -\- i 



0 



1 / - 3« e I qi e -, 



or, according to (168), 



= — 2^- e +2?^ 

 Introducing here the integralsinus and the integralcosinus by (169), we get 



2^ (p(g') = ci g\e — e j — i s\ q\e e j , 



or 



(171) ([>(</) = ci 5 sin g — si g cos g. 



To obtain 'L'(g) we observe that, according to the definition (169), 



d si X sin x 



doc 00 



d ci X cos X 



dtJC oc 



so that 



(172) i^'{q) = ci q cos g -]- si g sin g. 



Using known developments and tables for ci x and si x we are now able to 

 compute '|j and t}j' and hence also T. 



It would liave been possible to use the differential equation (166) 



im 'è + y-ï = 



in order to find the value of . This method has been used by Schlömilch (Grelles 

 Journal Bd 33 (1846)) who thus reduces the function to the integralsinus and the 

 integralcosinus. 



If the right member of (166) is put = 0, we first get the integral 



(173) y = 2^ cos x + ^2 sin x. 

 Varying now and we get 



(174) — ^, sin .r + cos x 

 if the condition 



(175) ^ cos r + ^ sin a; = 0 



dx dt 



is introduced. 



Lunds Univoisitets A is.sk rift. N. P. Avd. 2. Bd 28. 10 



