(a) 



586 Stoney — Cause of Douhle Lines in Spectra. 



Here the motion is represented by 



x = F,{t), 



y = F,{t), 



where F^ and F^ may be any two functions. By Fourier's theorem, these become 



X = Aq + Ai cos 6it + A2 cos ^2^ + • • • "^ 



+ Bi sin 0it + B2 sin O^t + . . . 



y - C^-\- Ci cos 6 J; + Ci cos d^t -^^ . . . 



+ Di sin 6it + Z>2 sin ^2^ + • • • 



(^) 



where ^1 = 27r»«i/?^, Bi='iTTiiuJT^ &c., in which m^ m2, &c., are positive integers 

 when Tis the periodic time (if any) of the motion, and in which nii m^, &c., are 

 numbers of which some at least are fractions when T is not the periodic time. If 

 the motion resolves itself into a finite number of terms, and if it is at the same time 

 one which does not repeat itself in a period however long, some of the numbers 

 nil 1^2, &c., are incommensurable. The coefficients (the A^s, B^s, C's, and i^'s), are 

 in all cases represented by the well-known definite integrals of Fourier's theorem, 

 and in some cases calculable from them. It should be borne in mind that a reso- 

 lution effected by Fourier's theorem is unique : in every case one such resolution 

 exists, and only one. 



We shall now proceed to prove that the four terms in these series which stand 

 in any one of the vertical columns of equations (b) represent a pendulous elliptic 

 motion ; so that equations (b) in effect resolve the original motion of equations (a), 

 whatever be its law, into partials, each of which is a pendulous elliptic motion. 



Take any vertical column, e.g. the k"" — 



Xf, = At, cos Q^f; 

 -f Bk sin Qyt^ 

 i/k = C4 cos 6kt 

 + D], sin 6 J, 



or, leaving the suffixes to be understood, 



X = A co^ 6t + B sin 6t, 

 y = C cos 6t + D sin 6i, 



and let us try whether we can identify it with the pendulous elliptic motion 



x' = a cos (6t + a), \ 



y' - b sin {Bt -\- a), j 

 in the same plane, and of the same frequency. 



{c) 



(^0 



{d) 



