574 Stoney — Cause of Double Lines in Spectra. 



The invariable line is a fixed line perpendicular to plane A, round which, plane B 

 is to be regarded as rotating with a swiftness cj = 2ttJcIj, in which k is the 

 frequency of this motion. 



The apsidal motion has already resolved the original elliptic motion into two 

 circular motions in plane B, viz., 



X 1 = + s cos St, \ 

 Zj = + s sin St, ) 



and X2--{-dco^Dt,\ 



(45) 

 Y^ = -d sin Dt,) 



which are the same as equations (3 a) and (3 h) when for brevity we write s and 

 d for (ff + 5)/2 and (« — J)/2, and S and D for 6 + ^, and 6 — ip. 



Draw three fixed axes : Os along the invariable line, 0.r along the direction at 

 which the intersection of planes A and B arrives at the instant t, and % perpen- 

 dicular to Ox in plane A. Then if a be the angle between the planes A and B, 



equations (4 a) furnish 



X = s cos St (5 a) 



along the intersection of planes A and B at the instant t, 



y = s sin St. cos a (5 5) 



along a line in plane A which is perpendicular to the intersection of A and B at the 



instant t, and 



s = s sin aS'^. sin a (5 c) 



along the invariable line. 



Equations (5 a) and (5 b) are an elliptic motion of P in plane -4, and when 

 affected by the precessional motion cat (where w = 2tt1cIj), furnish the circular 

 motions 



and 



Xi = -\- s cos- ^ . cos {S + o))t 

 Yi = + s cos^ ^ . sin (S + (o)t 



X2 = + s sin^ - . cos (S — a))t 

 y, = — 5 sin^ jr . sin (^S — (t))t 



{6 a) 



i6b) 



equations (5 c), (6 a), and (6 b) represent the whole effect ; (5 c) is a rectilinear 

 vibration of P perpendicular to the invariable plane, and (6 a) and (6 b) are two 



