VIBRATORY MOTION OF AN ELASTIC MEDIUM. 523 



If, however, we suppose that the vibrations of a polarized ray are in the plane of polarization, 

 we may shew as above, that 



B, = 5; = «% 

 B, = B! = b\ 



B, = BJ = c", 

 and therefore (12) becomes, 



or -^= -Z)B.(a'aAa+6'^ A/3 + d'y 1^^)01^ .v (14). 



28. Taking the equation (l.'S), we shall now find under wliat circumstances the force -— 



is in the direction of vibration. 



Let us choose a^ /3, y^ as in Art. 10, a^ being the direction of propagation, and 3, that of 



vibration ; and let « = >7,/3^. Then, as in the article just referred to, we have 19 = a^rf„. 



d^v . . . „ cf'J rf'f 



Now, the condition that the force — - may be in the direction B, is Ai/ . — -„ = (for — - is 



de •' ^' ^- df df 



already perpendicular to a^, and this condition makes it perpendicular to y^ likewise), or by (1.3), 

 A7,.(Z>a/.(o'aAa + b^(iA(i+ c^yAy)(3^ = 0. 

 But, by the general proportions of the notation D and A, we have A7 .Da ■ = A/3,., and 

 therefore A 7, . (Z>a, . )° = A/:}, . Da, ■ = - A7, . . Hence this condition becomes 



«-(Aa . 7,) (Aa . Si) + 6-(A/3 . 7,) (A/3 . /3,) + c^(A7 . 7,) (A7 . /3,) = 0. 



This is the well-known condition of Fresnel that the force brought into play by a transverse 

 vibration may be in the direction of that vibration; for 



Aa.7, = cos {ay) Aa./3, = cos (a/3,) &c. &c. 



To find the velocity of propagation in this case, we have, performing the operation A/3, on 

 both sides of (13), 



^' = {a^(Aa . /3/ + &^(A/3 . /3,)= + c= (A7 . /3,)=} <v, , 



and therefore the square of the velocity of propagation is 



a'(Aa. /3,)= + 6=(A/3. /3,)^ + c'(A7./3,)^ 

 which is FresnePs expression. 



29. We may treat the equation (14) in exactly the same way. 



M. O'BRIEN. 

 L'ppEU NoHwooD, April, 1847. 



