522 Mr. O'BRIEN, ON THE SYMBOLICAL EQUATION OF 



be true of V, and therefore Ai3 • V must be always zero. Consequently AID . V must be zero in 

 tbe particular case where ^ = 0, and ^ and tj are functions of .c and y only, in which case we have 



d.vdy \ ^dy ' dx 



. d,^ dv dr, 



observme that ^ h a — ^ ( = ol = 0. 



^ d.v dy dz^ ' 



That this expression should be always zero evidently requires E^ and E^ to be each zero ; and 

 in the same way we may shew that the other E''s are each zero. 



Hence, it appears, that the forces brought into play by transverse vibrations are not perpen- 

 dicular to the direction of propagation, except the conditions (g) hold. 



These conditions are therefore essential to the truth of Fresnel's Theory of Transverse Vibrations. 



26. Hence it follows that (12) is the most general form of the equation of vibratory motion, 

 when the transmission of a wave of transverse vibrations through the medium in every direction is 

 possible. 



27. Experiment shews that the six constants involved in (12) are reducible to three in the case 

 of ordinary Biaxal Crystals ; for it appears, that, when the plane of polarization of a ray coincides 

 with the plane a/3, the velocity of transmission in the direction a is the same as that in the direction 

 /3. Now, first, let us assume with Fresnel, that the vibrations are perpendicular to the plane of ' 

 polarization (a^) ; then, for the directions a and /3 the equation (12) becomes, in each case 

 respectively, 



dt' dy ^ dy ' df ' df 



Hence, if c denote the common velocity of transmission in the two directions, we have 



B, = b; = c\ 



In exactly the same way we may shew, that 



fi, = 5,' = a-, 



s, = 5; = b\ 



where a is the velocity of transmission in the directions /3 and 7 of a ray polarized in the plane 

 (iy, and b the velocity in the directions y and a of a ray polarized in the plane ya. 

 Hence the equation (12) becomes 



(Pv ^ , f d ^ d\ ^ / d d\ t^d d\„, 



-.Dl^M[y -^£)l.b^[a--y£),.c'[fi--.-^-^K\, 



dy dzl V dz d.v) \ d.v dyl 



...(13). 



or ^ = - {Dm . y(a'^a + fe%/3 + c'^y), 

 air 



or ^= - (Z>B.)-(a=aAa + 6^/3 A/3 + c=7A7)w, 



