578 Stoney — Cause of Double Lines in Bpectra. 



determine these latter. The middle line has the position which would be occupied 

 if there were no processional motion. 



The next matter to be considered is the effect of a periodic change in the 

 inclination of the two planes. Hence — 



Problem IV. — In what way will the spectrum of the gas be affected if there 

 be a periodic change in the inclination of the plane of the ellipse to the invariable 

 plane ? 



This problem is to be investigated exactly in the same way as Problem VI., which 

 is dealt with a few pages farther on. Since the angle a in equations (5 c), (6 a), and 

 (6 J) undergoes a periodic fluctuation, we are to write (^+ A sin 77^) instead of a in those 

 equations, ^, h^ and 17 being constants. If after making this substitution we apply 

 to the equations the same method of treatment as in Problem VI., we shall find 

 that the effect of the perturbation is to render the lines winged. 



Problem V. — What effect on the spectrum will a periodic change of ellipticity 

 have ? 



The change of ellipticity may take place in either of two ways : in one the 

 orbit will pass through a rectilinear form ; in the other it will pass through a 

 circular form. 



I. — To represent a change of the first kind we must substitute for a and h 

 of equations (1) the following: — 



{r cos e^) for «, 



(r sin i.t) for 5, 



where e = 27re/y, e being the frequency of the periodic change of ellipticity. 

 We thus get instead of equations (1) 



X = r . cos e^ . cos Ot, 

 y = r . sin d . sin 6t, 



which treated as in Problem II. give 



r 



^ = 2 



^ = 2 



C0s{e-e)t+C0s{6+e)t 



G0s{6-e)t - cos(^+ e)t 



These equations represent two rectilinear motions at right angles to one another, 

 of frequencies m + e and m — e, and of equal intensity. They, accordingly, 

 would give rise to a pair of equal lines in the spectrum of the gas. If there be 

 absidal motion also, each of these will be doubled, and two pairs of equal lines will 

 present themselves. 



