Stonet — Came of Double Lines in Spectra. 581 



This substitution being made in any of the equations of Problem II. , suppose in 

 equations (3 a), we get 



X, = ^^ cos [{9 + y\i)t + a sin ^f], 



^ h ^ ^^^^ 



Y, = "^ sin 1(0 + xlj)t+a sin Cf\. 



To see how this will operate, imagine /3 to be the value through which i,t passes 

 at the instant when t = t. Then for a short period of time after 



sin Ct = sin /3 + cos fi .d .t,t 



q 

 = sin ;8 + cos ^ .2tt^ dt, 

 J 



in which dt is to be regarded as equal to ^ — r for a short time after the epoch t. 

 Putting this into (14) we find that equations (14) during a short period fm'nish a line 

 of frequency (m + n+ a cos /3 . q). By dividing j/q the periodic time of ^Z^, into equal 

 parts ; by giving to a series of yS's the values which l,t has at the commencement 

 of each of these equal intervals of time ; and by then supposing that the duration 

 of these intervals decreases while their number increases indefinitely : we find 

 that the total effect is the limit (when N increases indefinitely) of a band of N 

 lines of equal brightness, crowded towards the middle, and becoming more and 

 more spaced asunder towards the edge — in other words, it is a nebulous or ' diffuse ' 

 line fading out equally* on both sides. The middle of the line has the frequency 

 m + n, and its wings extend from m -^^ n + aq on the more refrangible side, to 

 m -\- n — aq om the less refrangible side. 



The same appearance in the spectrum would result from a periodic oscillation 

 affecting either of the other pertm'bations ; and in Problem IV. we have found that 

 wings will present themselves if there is a fluctuation in the inclination of the plane 

 of the ellipse to the invariable plane. Accordingly, nutation makes the lines 

 diffuse, and a fluctuating inclination makes them winged. 



* That is, equally, if the nutation is a mere pendulous one. 



4 N 2 



