Excerpt from The Collected Mathematical Papers of James Joseph Sylvester, Vol. 1
The following is, I believe, the first successful attempt to obtain the full development of Fresnel's Theory of Crystals by direct geometrical methods. Hitherto little has been done beyond finding and investigating the properties of the wave surface, a subject certainly curious and interesting, but not of chief importance for ordinary practical purposes. Mr Kelland, in a most valuable contribution to the Cambridge Philosophical Transactions, has incidentally obtained the difference of the squares of the velocities of a plane front in terms of the angles made by it with the optic axes. I have obtained each of the velocities separately, and in a form precisely the same for biaxal as for uniaxal crystals.
I have also assigned in my last proposition the place of the lines of vibration in terms of the like quantities, and that in a shape remarkably convenient for determining the plane of polarization when the ray is given. For at first sight there appears to be some ambiguity in selecting which of the two lines of vibration is to be chosen when the front is known.
Without the most careful attention to preserve pure symmetry, the expressions could never have been reduced to their present simple forms.
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