VIBRATORY MOTION OF AN ELASTIC MEDIUM. 511 



When the six conditions above alluded to are introduced, the equation of motion for a crys- 

 tallized medium becomes 



drv I d ^ d , d \ _- 



i 



Where .4, A.;, A, are the three coefficients of direct elasticity with reference to the three axes 

 of symmetry, and 5, Bi B^ B^ B^ BJ the six coefficients of lateral elasticity with reference to 

 the same axes. 



If the vibrations be transverse, this equation is reducible to the form 



^ = - {D-D.y{a'^a + b'r,^ + c'^y) 



or — = -{2>i3.)=(a'*aA« + 6-/3A/3 + c'7A7)«, (A), 



dt 



assuming the vibrations of a polarized ray to be perpendicular to the plane of polarization. 



The well-known condition that a plane polarized ray may be transmissible without subdivision, 

 and the expression for the velocity of propagation, may be immediately deduced from this equation. 



If we assume the vibrations of a polarized ray to be in the plane of polarization, the equation 

 becomes 



^ = - Z?5B.(a^aAa-i-ft'/3A/3 + 0=7^7)2)19.1' (5). 



The equation {A) agrees in all respects with Fresnel's Theory, and the equation {B) includes 

 Professor Mac Cullagh's three equations. It is curious that {A) and (B) should differ from each 

 other only in the order of the operations performed on v in the second member. 



Investigation of the Symbolical Equation of Vibratory Motion of an UncrystaUized 



Medium. 



1. Let ii( = aw -(- ^y + yz)* be the symbol of any particle (P) of an elastic medium in n 

 state of equilibrium, v(^ = al^ + (irj + yX.) the symbol of the displacement of the particle at any 

 time t, u + cu (du — a^x + (iSy + ySx) the symbol of the equilibrium position of a contiguous 

 particle (P), and v + Sv (Sv = aS^ + (i^rj + ySi^) the symbol of the displacement of P' at the 

 time t ; then we have 



, dv . dv 5 dv . tf « d'v 



ov = -— d.v + -J- dy + —- 6x + -L -— o,v + -— - 

 d.r ay dz dou daidy 



This equation expresses the disarrangement, or state of dis[)latenienl, of the n)edium in the 

 immediate vicinity of P, for cu is the relative displacement of P' with reference to P, and by 

 giving different values to o.v oy Iz in (1), corresponding to the different particles near P, we (ind 

 the dih])lacements of those particles relatively to P. 



2. In consequence of the disarrangement of the medium in tlie vicinity of /', repre.senteii by (1), 

 a force will be brought into ))lay upon /•"; our oliject in tojind this force. 



Vol.. VIII. I'akt IV. 3U 



av . aw 5 «u „ a V ^ o u . „ 



;j-^ d.r -t- — riy -I- -j-da; H- ^ — ^ o,v + -^^3- hmdy + &c (1). 



