Stoney — Cause of Double Lines in Spectra. 579 



II. — To represent a periodic change of ellipticity, in which the orbit passes 

 through a circular form, we must substitute in equations (1) 



(r + p cos e^) for a, 



(r ~ p cos e^) for b, 

 where e = STre/r. We thus obtain 



X = [r + p cos et) cos 6t, 



y — {r — p cos et) sin 6t, 

 which are equivalent to 



z = r cos 6'i; + ^ cos (^ + e) j; + ^ cos {6 - e) t, 



y = r sin Qt -^s\.n{e ^- e)t -^ sin {6 - e) t. 



This motion, if p is small, produces a line at frequency m, with two equal satellites at 

 frequencies m + e and m — e, i. e. one on either side of the primary. 



Another perturbation which may possibly present itself would consist in the 

 alternate contraction and dilatation of the ellipse. This is represented by the 

 equations 



X = a . cos et . cos 6t, \ 



(13 a) 

 y = b . cos e^ . sin 6t, ) 



where 6 = 2Trm/j and e = 2Tre/j. The energy of this motion is (a' + P)/2, if we 

 represent the energy of the simple elliptic motion of equations (1) by a" + P. 



Problem VI. — What appearance in the spectrum would this perturbation 

 occasion : 1°, if alone ; 2°, if accompanied by an apsidal shifting of the ellipse ? 

 1°. Equations (13 a) are equivalent to 



a; = - , cos (^ + fc) i^ + 5 cos (6 — e)t, ] 



b h (13^) 



^ = ^.sin(^+€)if+|sin(^-e)i!. ) ^ 



Hence the perturbation, when alone, occasions two equal lines of intensity 

 (a^ + ^^)/4, at the positions m + e and m — e on a map of oscillation-frequencies. 

 2°. Equations (13 5) are equivalent to 



X "^ X\ -J- ^2? 



y = yi + y2, 



a „\ ( a 



where 



Xi = -^ cos St 

 y, = ^ sm St 



and 



3^2 = 5 cos Dt, 

 y2=2 si^ ^^^ 



TRANS. EOT. DUii. SOC, N.S. VOL. IV., PART XI. 4 N 



