Kinematical Analysis of B 'aimer } s Formula. 21 



3. A Kinematical Analysis of Balmer's Formula. 

 Under this title in section 3 of " The Cause o£ the 

 Structure of Spectra " an attempt was made to trace Balmer's 

 formula to certain simple effects of relative motion. In that 

 paper quite a number of numerical laws or indications of 

 such laws are shown to imply the prominence of kinematical 

 considerations. Yet since Rayleigh suggested in 1897 that 

 Balmer's formula points to kinematic, rather than dynamic, 

 relations, many elaborate attempts have been made to find a 

 dynamical explanation of the relations amongst the lines in 

 a spectral series. To follow up the dynamical theory here 

 advanced for the fundamental constant of atomic vibration 

 in Balmer's formula it seems desirable to make my former 

 kinematical analysis of Balmer's formula more definite in 

 the following way. Consider a vibrating circle one point of 

 which is constrained to be a node. The circumference of the 

 circle is half a wave-length. The stationary wave is the re- 

 sultant of two trains of waves which are travelling in opposite 

 directions round the circle, and are always being reflected 

 at the node. Let the angular velocity of each train be co. 

 Suppose now that the node A is caused to move with angular 



Fig. 3. 



velocity 12 because the constraint producing it so moves. 

 Then relatively to A the one train is moving with angular 

 velocity co — 12, and the other with — co— 12. Thus the con- 

 straint* will be met by the one train after time 2ir/(co — 12), 

 and by the other after time 27r/(a> + X2), that is to say, the 

 half wave-length of the one train which was originally 27rc 

 is changed to 27rca>/(o>-fl2), and that of the other to 

 27rcft)/(o) — 12). But upon reflexion each of these altered 

 half waves is followed by the other. Thus each whole wave 

 X of one train is replaced by two parts A.g)/2(o) + 12) and 

 \co/2(co— 12). The state of affairs is represented graphically 

 in the diagram where AB=AC+CB and represents part of 

 the circumference of the circle developed as an infinite 



