IN CRYSTALLIZED MEDIA. 133 



The two last equations give 



cJ^Tz = ~^~^ = bx-ay = w SU PP° se - 



Hence the last of the equations (9) becomes 



o? = Lx' 2 + My' 2 + N%\ 



But 



x' 2 + y' 2 + z' 2 = w 2 {(cy - b*y + (az - ex) 2 + (bx - ay) 2 \, 

 m a, 2 {(b 2 + a 2 ) z 2 + (c 2 + a 2 ) f + (// + v 2 ) of -2 (bcy% + abxy + acx%)\, 

 = w 2 {(a 2 +b 2 +c 2 )(x 2 +y 2 + « 2 ) - {am + by + c%)* }, 

 - w 2 (« c + y 2 + z 2 ) = x 1 + f + » 2 , 



.: w 2 = 1, 

 and our equation finally becomes 



1 = Lx' 2 + My 2 + Nz' 2 . (io.) 



We thus see, that if we conceive a section made in the ellipsoid to 

 which the equation (10) belongs, by a plane passing through its centre 

 and parallel to the wave's front, this section, when turned 90 degrees 

 in its own plane, will coincide with a similar section of the ellipsoid 

 to which the equation (8) belongs, and which gives the directions of 

 the disturbance that will cause a plane wave to propagate itself with- 

 out subdivision, and the velocity of propagation parallel to its own 

 front. The change of position here made in the elliptical section, is 

 evidently equivalent to supposing the actual disturbances of the ethereal 

 particles to be parallel to the plane usually denominated the plane of 

 polarization. 



This hypothesis, at first advanced by M. Cauchy, has since been 

 adopted by several philosophers; and it seems worthy of remark, that 

 if we suppose an elastic medium free from all extraneous pressure, we 

 have merely to suppose it so constituted that two of the wave- dis- 

 turbances shall be accurately in the wave's front, agreeably to Fresnel's 



