Stoney — Cause of Double Lines in Spectra. 



587 



Let fi be the angle between the axes Ox and Ox'. Then the motion {d) referred 

 to the axes Ox and Oy becomes 



X = a cos {6t + a) cos fi — b sin (6t + a) sin ^8, 



y = a cos [dt + a) sin )S + J sin (^^ + a) cos y8, 



which when expanded becomes 



X = cos 6t (a cos a cos /3 — J sin a sin /S) ^ 



— sin 6t (a sin a cos yS + 5 cos a sin yS), 

 F = cos 6t (a cos a sin JS + b sin a cos yS) 



— sin 6i (a sin a sin ft — b sin a cos ^). 



Now, we can determine a, h, a and /3, so as to make 



a cos a cos /3 — b sin a sin ft = + A,"^ 

 a sin a cos ft + b cos a sin ft = — B, 

 a cos a sin ft + b sin a cos ft = + C, 

 a sin a sin ;8 — 5 cos a cos ft = — D. 



(^) 



(/) 



Hence, when a, b, a and ft, have the values so determined, the pendulous 

 motion represented by (d) is identical with the motion represented by (c^). 

 Hence the theorem corresponding to Fourier's theorem is — 



Theorem A. — Any motion of a point in a plane may be regarded as the 

 coexistence and superposition of definite partials which are the pen- 

 dulous elliptic motions determined as above, one from each of the 

 several vertical columns of equations (b). 



These elliptic partials will all be in the plane of the original motion. They 

 will, however, in general lie in different azimuths in that plane, and be in different 

 phases at any one time.* 



What lends importance to this theorem is that the resolution effected by it 

 in our calculations is identical with that which an undulation of electro-magnetic 

 stresses in the open aether (as, for example, the great complex undulation which 

 reaches om- atmosphere from the sun or a star) does actually undergo when the 



* In order to characterize the kind of motioa which takes phice in a partial, it is suificient to deter- 

 mine three constants, viz. a, h, and 6 (the axes of the ellipse and the swiftness of the motion in it). But 

 to determine the position of P at each instant, it is necessary to determine two more constants /3 and a 

 (jS, the azimuth of the ellipse in its plane, and a the position of JP in it at the instant t = 0). 



A continuous spectrum arises when the 6's of the partials are indefinitely close, a spectrum of lines when 

 they are at intervals that can be perceived. 



TEANS. EOr. DUB. SOC, N.S. VOL. IV., PARI XI. 4 



