568 Prof. 0. W. Richardson on a Theory of the 



by parts n — 2 times we obtain 



. /r«fo /";"(» -l) if 



cos^ro + y?'! — (/3 + 7>+ — ^— tt 



(/3 + y) tt -V 



?z + 1 

 cos/3r — yn — (/3 — y)r : ' 



(/3-7) ?l -V 





W + 1 



-m 



(/3 + y) n - 2 r n ^ 



cos ftr — yn — (/3 — 7) r H .r— 7r 



(^-7) n ~V- 1 



)*..}< 21 > 



72 -I- X 7 



»=« (72-1) J / cos 7( r i~ r o) + — 2~ 7 cos y(r — n) + - 



»=3 2r w y~ — (£ + ry)»-2 (fi-yT~ 2 



n + 1 



IT 



nlC ri / cos / 5r o + T'i-(/3 + 7>+^— 7r 



¥ i r 



r \ [j3-\-y) n - 2 r n+1 



cos /3r — jr l — (/3 — y)r H — 7r 



)*■]. 



(^-^-2^+1 / J. (22) 



Confining our attention for the moment to the integrated 

 part of equation (22), we see that all the terms which involve 



n + 1 



cos 7 (r —r 1 )H - — 77-, and therefore represent vibrations of 



L 



the frequency of the free vibrations of B, depend only on r 

 the initial position of B. The disturbance represented by 

 these terms will therefore rapidly disappear if there is any 

 damping ; whilst even if they persisted their average effect 

 in producing a displacement of the lines emitted by A would 

 be zero on account of the difference in the period and the 

 uncertainty of the phase. The only terms which are of 



