210 



The Rev. M. O'Brien on the Propagation 



posed transversal, and the particles of matter absolutely 

 fixed. 



The particles of matter being supposed absolutely fixed, 

 the equations (2.) will be identically zero, and the part under 

 the sign S in (1.) will also be zero; hence we shall have 



d- « , (t^a. „ /d^ OL d- a\ ~] 



+ (A-B)(/f + ^-) 

 \dxd y d X dz) 



d- /3 d^ |3n 



c« 



(1-) 



d'^^^d^^ 



dt 



dy' ^ \dx^ ^ dz^) 



+ (A - B) (^ + /-f ) 



\dx dy dy d z / 



C/3 



df' -" dz' "^ ^\dx' "^ 



+ (A 



Xdxd: 



+ 



dy^J 



d^^ 

 dy d'. 



)-c,J 



(2.) 



(3.) 



Let the equation to any one of the planes of like 

 phase be px-\-qy-\-sz = u (4.) 



When jJQ s are the cosines of the angles it makes with the 

 coordinate planes respectively, and therefore n the perpen- 

 dicular upon it from the origin. Now it is evident that « /S y 

 must be functions of ii alone, so far as x yz are concerned, 

 otherwise we should have particles in the same plane of like 

 phase in different states of vibration, which is absurd. 

 Again, if v be the constant velocity of propagation, we must 

 have 



d^u _ ^ d^u d^ _ ^d^ d^y - 2^^y 



dt~ ~ ""'dii^' dt' ~ " du'' dt 



= "'^' ••■•('•) 



For if we suppose u and t always to vary in such a manner 

 that a /3 y remain invariable, then —- obtained on such a sup- 

 position will be the velocity of propagation (o). Now, on this 

 supposition we have </ « = 0, that is. 



d'^a 

 du'' 



