Displacement of Spectral Lines produced by Pressure. 567 



by vf/. At the instant £ = B is distant r from A, after 

 which it moves directly towards A with uniform velocity v 

 until a time t± has elapsed when it is distant i\ from A. Let 

 r be the coordinate of B at time t, and let the direction of £ 

 coincide with the line of motion ; then the electric force due 



2^Cn COS 7)t 



to A acting on the electron at B at time t = — 9 — ^— 



so 



that the equation for yjr is 



d 2 \/r 2e% cos pt 



m ~ 



t, • • • (17) 



where \ is the coefficient which determines the natural period 

 of -\jr (see equations 2 and 3). In this the Doppler effect is 

 disregarded as being small, whilst the reaction of B on A 

 and the decay of f with time have been avoided artificially. 

 r is evidently connected with t by the relation 



r = r — vt. ...... (18) 



By changing the independent variable from t to r the differ- 

 ential equation (17) can readily be solved, leading to 



. /Vffo P ri sin /3r + 7 n - (ff + 7 )r-sin fao-y^ - (fi-y) r j r ^v 



r V m v Jr Q r z y 



where /3 = p/v and <y=e/v^/\m (20) 



This is the particular integral only ; the complete solution 

 contains in addition the terms C sin y(r — r t ) + D cos y(r — r,). 

 These terms represent vibrations of the frequency of the 

 natural vibrations of the electron in the atom B due to 

 the initial impulse and may be left out of account in the 

 present case. The integral (19) may be evaluated by 

 integration by parts. After two successive integrations of 

 the denominator, the unintegratecl part contains as a factor 

 one of the integrals 



Jsinu _ rcosw 

 au or 1 

 u J u 



du. 



These can be obtained from tables, but as the object of this 

 investigation is to find the relation between the solutions for 

 the two cases when the atom B is moving and when it is 

 fixed, it is more convenient to proceed by integrating the 

 numerator and differentiating the denominator. Integrating 



