212 The Rev. M. O'Brien on I lie Propagalion 



we may show that 



/3 = 6 cos k {vi—u) 

 y = c cos k [yt — u). 

 The following two consequences follow from these results : 

 1st, That the particles must necessarily vibrate according to 

 the cycloidal law; a plane wave cannot be transmitted with 

 a uniform velocity unless this condition hold. 



2ndly, That the velocity of transmission depends on the 

 length of the wave; for if A be the length of the wave, 



o C 



k = —; and .-. since F = -^ — rr, we have 

 X o" — r> 



.2 = B - ~ x\ 



4 7t^ 



The second case we shall apply our equations to is that in 

 which all the circumstances are the same as in the preceding 

 case, except that the vibrations are normal. If this be the 

 case, instead of the condition (6.) we have the conditions 



_a _ _^ _ y_ 



P ^ « ' 



and therefore (7.) becomes 



or („2_A)^^ + C« = 0. 



Hence the same consequences follow in this as in the 

 former case ; the only difference is, that instead of k" = 2"^^ 



Q 



we have k'^ = —3 r-, and therefore 



u^ — A 



u^ = A- -^,x2. 



Hence it follows that transverse and normal vibrations are 

 propagated with different velocities, viz. 



/ B — 7~~^^^ ^"^ \/ ^ r~2^'^ respectively. 



It is evident that if C were zero equation (7.) would be sa- 

 tisfied quite independently of a by putting u^ = B if the vi- 

 brations be transverse, or A if normal. Hence it follows that if 

 the particles of matter did not act on the particles of tether (i. e. 

 if C was zero), a plane wave might be transmitted with a uni- 

 ibrm velocity whether the law of vibration was cycloidal or not. 



