By Professor Pell. 21 



These equations giye 



2 sin n 9 sin \ Q 



— r-fl Xj^ = X x^, = \ a cos a f 



cos -| 9 "^ 



^ __ cos (n—r + i) 5 



cos i 5 

 X « cos (n — r + 1) 9 cos jx ^ 



.(8) 



2 sin w 9 sin i 3 

 and tlie relative displacement is 



X a sin (w — r) & cos jw, t 

 ^ ' sm W S 



Let [J, = 2 oil sin \I/, then 9, operating upon cos ju,i5, is equal to 

 2 v^, and 



A « sin 2 (n — r) \|/ cos [x, i 



Now X, representing the ratio of the action of an atom of ether 

 to the mutual action of the atoms of a solid body, is very small, 

 almost infinitesimal. The above coefficient of cos [j.t is therefore 

 wholly insignificant except when sin 2?z\J; is very small, or when 

 2 71 ^ = SL multiple of tt. In this case, as before explained, the, 

 quantity by which X is multiplied is large, but not infinite : for 

 as the amplitude of the vibration increases, the time of vibra- 

 tion is slightly increased, as indicated by the factor c, introduced 

 in the second approximation ; and \J/ is thus slightly diminished. 

 The only values of vf; which produce any sensible efi"ect are 



— , _, — , &c Let vl/ = — = u Yj where u is an 



2 n 2 n 2n ^2 n ^ 



X a 

 integer; puttmg h = ^ — -. ., we have, supposing the diatoms 



to have been initially at rest, in their positions of stable 

 equilibrium, 



„ . h cos (2 n — 2 r + 1) y 



^r = ^J^ + 2 rCfg cos fJist ^: ; !_' COS jU,^ 



sin u y 

 where Oi r^a are arbitrary constants, and ^^ = 2 m sin s y. 

 Putting t = 



= «, + 2 A - ^cos(2^.-2r + l)vy 



sin y y 

 Equation (8) gives 



cos (2n ~ 2r + 1) sy 



cos s y 



