252 Dr. J. W. Nicholson on the Number of 



Reduction of the Integrals, 



Writing q\ T2 = co in the first result, we obtain 



vk C " coda) 



. __. I £~ w . 



A ^°"o Jo 0)Jra 

 so that this ratio does not involve q explicitly, If co = — a + f , 



or 



v«==\Co- (l-ae tt Ei(- a )) 5 • • ■ ( 27 ) 



where E/( — a) is the exponential integral function defined by 

 Bi(-«)=f "*«"«, .... (28) 



of which exhaustive tables have been constructed by Glaisher*. 

 If a is not small, we may write asymptotically 



E*-(-«)=^g44-J...), • • (29) 



the series being formally divergent after a certain number 

 of terms, after the usual manner. Taking the first two 

 terms only, as is sufficient for the good conductors, we 

 derive 



^=\Co- /a==N 2 \V/7r 3 m 2 C(7o, . . . (30) 



this formula being then more accurate than the experimental 

 measures with which it is to be associated. 



Under the same circumstances we may expand the second 

 formula, which yields 



^K-lVoV^CT^-^I-^h-^...), . . . (31) 



or for good conductors, 



^-K 2 = K~-2<ro ^^OTm/a=K-7rmC 2 o-o7« e 2 N, (32) 

 quoting again the value of a in (22), 



* Phil. Trans. 160 A (1870). An abridged form is given in Jalmke 

 k Erade, Fiinctionentafeln, Leipzig, 1909, 



