260 Mr. Tovey on the Optical Theo7y of Crystals. 



Vif must, as I have shown, be a maximum or minimum, these 

 equations give 



a} dx^ ^'if dxf -^^ z^ dz^ = , 

 Idx' + m dy' -^ ndz^ = . 

 From the first and second of these equations we find 

 (c"^--c"-) x' do^ + {c"'-^&) 7jdyf = ; 

 and from the second and third 



{n x' —l ss') d x' + {nif—mz^) dy^ = 0. 

 Combining these two, we have 



(c^'-r).{7ix^-lz')y^-[c"^^c').(ny'---mz^)x' = 0, 

 which, by reduction, becomes 



y^ - ^'1 ^+{c^- c') 4- + (^'^- ^') 7 = 0- (*•) 



The equations (a) and (b) are virtually the same as those 

 which Mr. Sylvester has so denominated, and from them all 

 his other equations are derived. 



Between the results of my investigation and those of 

 Fresnel's there is a difference sufficient to afford a criterion 

 for deciding which of the two is correct. (See L. & E. Phil. 

 Mag., vol. ix. p. 429.) If Fresnel's be correct the vibrations 

 of rectilinearly polarized light must be perpendicular to the 

 pla?ie of polarization^ so that in the undulation constituting 

 the ordinary ray of an uniaxal crystal, the direction of the 

 vibrations must be perpendicular to \ts principal plane, (See 

 Airy's Tract on the Undulatory Theory, art. 100.) Whereas 

 if my investigation be correct, the vibrations constituting the 

 ordinary ray are parallel to the principal plane (L. & E. Phil. 

 Mag., vol. ix. p. 424,) and consequently the vibrations in 

 rectilinearly polarized light are parallel to the plane of polari- 

 zation. Now as my theory not only agrees in this result 

 with M. Cauchy's, (as I have remarked at p. 425, vol. ix.) but 

 is also confirmed by Professor MacCullagh's theory of cry- 

 stalline reflection, (L. & E. Phil. Mag., vol. x. p. 422,) I 

 feel persuaded that it is the true one. 



I am. Gentlemen, yours, &c. 



Littlemoor, Clitheroe, Feb. 6, 1838. JoHN ToVEY. 



P.S. In my paper in your January Number, p. 12, equa- 

 tions (7.), for n in both places, read n^; p. 13, line 21, for 

 (1— sin {n^t—Jc x)) read (1 — sin^ {n^t—kx)) ; and line 23, for 



