254 Mr. W. Sutherland on the 



The period r of the type o£ uneven vibration thus established is 



TS= WJ_ + _1\ . . (4 ) 



v U + w/v 1—ufvJ' w 



and on comparing this with (3) we get the relations 



r ~ v^ ~ ?> W 



U = J*-^-, (6) 



whereby \/V, which may be denoted by r , appears as the 

 natural period of vibration of the circle when the reference- 

 vector is at rest, and also the ratio of the two velocities v 

 and u appears to have a series of values depending on m + u w 

 It may be noticed that this explanation of the origin of 

 Balmer's formula is kinematical, in accordance with Rayleigh's 

 surmise that that formula must indicate kinematic rather than 

 dynamic relations (Phil. Mag. [5] xliv. 1897). Now, as we 

 have already seen, the fundamental period of the simple 

 harmonic vibrations of a circle being taken as 1, its possible 

 periods of vibration are given by l/(r±p/s), and if we derive 

 these harmonic motions by projecting the motions of the ends 

 of uniformly revolving radius-vectors on fixed diameters in 

 the nsual way, the relative values of the periods of revolution 

 of these vectors will be given by l/{r + p/s), so that we under- 

 stand equation (6) as giving us the ratio of such angular 

 velocities of vectors as we have just been considering. We 

 must postpone the further consideration of our formulae till 

 we have considered Rydberg's laws. 



4. Bydhercfs Laws. 

 In the Phil. Mag. [5] xxix. Rydberg gives a summary of 

 the results arrived at in his chief memoir, published in French 

 by the Swedish Academy (Svensk. Vet. AJcad. xxiii.). In the 

 first place he extends Hartley's discovery that certain lines 

 of the spectra can be grouped in pairs or in threes, such that 

 the difference or differences of their wave-numbers is or are 

 the same for all the pairs or threes, and makes this an 

 important principle in helping to pick out from the bewildering 

 number of lines those which form definite series^ because the 

 pairs or threes so characterized are members of parallel series. 

 If one of the series can be expressed by a formula 



« = tt„-B/(/» + /*) 2 , 



then in the case of pairs the other is given by 

 n=n Q + v — B/^-t-/*) 2 , 



