590 Stoney — Cause of Double Lines in Spectra. 



Thus the motion (C2) is equivalent to 



X2 = a cos {dt+€+ a) cos^-b sin (6t+c + a) sin )8, 



F2 = \aeos{dt+€ + a) sin^ + b sin {6t + e + a) cos jS] . cos w, 



^2 = [« cos (^^ + e + a) sin fi + b sin {6t+e + a) cos ^S] . sin co, , 



(^3) 



which we shall identify with the motion (b^) if we can determine a, b, a, /3, y, and 

 (o, so as to make the coefficients of cos{6t + e) and sin (6i + e) identical in equa- 

 tions (^5) and (cg). 



Now, the equations (c^) are identical with 



(c.) 



in which 



(rf) 



X2 = cos {6t + e).k - sin (^^ + e) •/>, 



Yi = [cos (^if + e) . g- - sin {6t + e) . r] . cos co, 



Zi = [cos (Qt-\- ^).q- sin (^^ + e) . r] . sin w, 



A" = a cos a cos ^ — b sin a sin ^, n^ 

 jt? = a sin a cos /S + 6 cos a sin /3, 

 ^ = a cos a sin /3 + J sin a cos yS, 

 r = a sin a sin ^ — b cos a cos jS. ■' 



Identifying the coefficients in (^5) and (C4) we find that the equations to be satisfied 



are — 



h — u cos y, (e,) 



JO = Z' sin y, (ej) 



5' cos oj = 2{ sin y, (fj) 



?• cos u> = — V cos y, (^4) 



g* sin 0) = M, [e^) 



r sin w = — N. {e^) 



M 



From (gj) and (ej), we find that 



tan &) = 



Similarly, from {e^ and (ee) we find that 



tan CO 



Equating these, we find that 



u sm y 



2^ cosy' 



(/O 



(/O 



tan y = ■^, 



which determines y. Having found y, equations ("ei) and (e^) determine k and jo, 

 and equation (/i) determines w ; and having found y and w, either equations (64) 



