Theory of Refraction in Gases. 473 



The equation of propagation of the waves is 

 1 9X BM . 





Hence we oet 



dx 

 of 



^ , -__^-v - ^^^1 ^2/ Jo -^'-P 



^ 3 ^ w^ _^ 



Now 



A' 

 Jo .y— ^2 CO^ ^ ^ Jo ^—P 



Hence, taking the principal value of the integral, we get 



3 W^ \?»1 ???2^ 



1-e ^-^ 



where 



nd 



f =!=--*<') 



EH^}=y+ilog.^'^-fS;°^, 

 and 7 is Eiiler's constant. 



(5) Digression on the Functions involved. 



The function Ei(.v) involved has been discussed in Lamb's 

 • Hydrodynamics/ p. 397, and by Schlomilch *. Kecently, 

 Mr. E. VY. Earnest has discussed the form of the asymptotic 

 expansion ; and I am indebted to him for considerable 

 assistance in examining the general form. 



I found it necessary to calculate the numerical value of 

 several functions involving Ej(a;). The results are given in 

 the following Table, in which E(<^) stands for xe~^''Eii{x). 



^ Crelle's Journal, t. xxxiii. p. 316. 

 t Phil. Trans, cxcix. p. 447 (1902). 



