IN THE INTERIOR OF TRANSPARENT BODIES. 423 



Now by giving proper values to v* and — in (1) and (2), we may 



satisfy both these equations, and consequently the six equations of 

 motion; hence the assumed values of a, /3, 7, a t , /3,, 7,, satisfy the 

 equations of motion provided the vibrations be transversal, and v* and 



— ' be so assumed as to satisfy (1) and (2). 



It is evident from the same reasoning as that employed in Art. 12., 

 that B'k 2 is small compared with C, also by Art. 14., a, is small com- 

 pared with a; hence, for a first approximation, omitting the terms 

 »»Cff,, and m,ITPa 4 , we obtain from the equation (1), 



v 2 = B + — '■ 7 - (k = — - , 

 m k 2 \ X / 



which is the result previously obtained. 



For a second approximation we must retain the term m^Ca,, but 

 we may omit m^'Tia,, and mB'k 2 a, in (1) and (2), and then we have 



It a = '-^ (a - a), 



v 2 — mB " 



subtracting the second of these equations from the first, and dividing 

 out a — a t , we find 



X 2 v 2 — mB v l - in i B i 



which is the same relation between v and X as that which I obtained 

 in the Philosophical Magazine, for March 1842, by a different method. 



A third approximation may be easily obtained by retaining the last 

 terms of (1) and (2) and eliminating — , which will give a still more 



tt 



exact relation between v and X. 



§ 30. If C = 0, k 8 divides out of (1) and (2), and therefore the 

 velocity of propagation has no dependance on the length of the wave. 



3 B2 



