Stoney — Cause of Double Lines in Spectra. 



571 



It will obviously give rise to a single line in the spectrum, whose position on a 

 map of oscillation-frequencies is w, and whose intensity may be represented by 

 a^ + h\ 



Peoblem II. — How will this simple spectrum be altered 

 if there is an apsidal motion of the ellipse in its own 

 plane ? 



Draw rectangular axes of co-ordinates from the centre of 

 the ellipse as origin, and at an angle t/*^ with the axes Ox and 

 Oy of equation (1). Regard the axes OX and OY as fixed, 

 and let i/f = '^tmlj. The ellipse will then travel round with an 

 apsidal motion such that n is the frequency of the apsidal 

 circuits in one jot of time. The co-ordinates of P referred to 

 the fixed axes are — 



X—a cos 6t . cos xpt—b sin Ot . sin xjji ; 



Y — a cos 6t . sin xpt+b sin 6t . cos tpt. 



equations which are equivalent to 



^^ a + b ,„ ,,, a — b 



X = -^r- cos (d + xli)i+ ~^- cos {6-xl,)t; 



Fig. 1. 



2 



a — b 



In other words, 



where 



r = ^ sin (6 + ^)t-^" sin {6 - xlj)t. 



X = Xi + X2, 



r= r, + Y„ 



Fi = + ^.sin(^-f VH^ 



(2) 



and 



X,^ + 



2 

 a — b 



(3 a) 



cos {6 — ^)t, 



Y.^-'Lj.sinid-lxl^y, 



(Sb) 



each of which represents a circular motion. Accordingly an elliptic motion whose 

 frequency is m, when affected by an apsidal perturbation whose frequency is n, is 

 equivalent to the motion of P resulting from the two circular motions represented 

 by equations (3 a) and (3 b). These circular motions are in opposite directions, 



TKANS. EOT. DUB. SOC, N.S. VOL. IV., PAKT XI. 4 M 



