Mechanical: Theory of Circular and Elliptic Polarization. 401 
which differ from the common equations of vibratory motion by the 
two additional terms containing third differential coefficients multi- 
plied by the same constant C, this constant having opposite signs in 
the two equations. The quantities § and y are, at any time 7, the dis- 
placements parallel to the axes of « and y, which are supposed to 
be the principal directions in the plane of the wave, one of them 
being therefore perpendicular to the axis of the crystal. The con- 
stants A and B are given by the expressions 
A=a, B=a*— (a — 6) sn’y, 
where a and 8 are the principal velocities of propagation, ordinary 
and extraordinary, and tf is the angle made by the wave-normal (or 
the direction of z) with the axis of the crystal. ‘The only new con- 
stant introduced is C, which, though the peculiar phenomena of 
quartz depend entirely on its existence, is almost inconceivably 
small; its value is determined in the paper just referred to. The 
equations are there proved to afford a strict geometrical representa- 
tion of the facts; not only connecting together all the laws disco- 
vered by the distinguished observers to whom M. Cauchy refers, and 
including the subsequent additions for which we are indebted to 
Mr. Airy, but leading to new results, one of which establishes a re- 
lation between two different classes of phenomena, and is verified 
by the experiments of M. Biot and Mr. Airy. Having, therefore, 
such conclusive proofs of the truth of these equations, we are en- 
titled to assume them as a standard whereby to judge of any theory ; 
so that any mechanical hypothesis which leads to results inconsistent 
with them may be at once rejected. 
Now I assert that the mechanical hypothesis of M. Cauchy con- 
tradicts these equations, and therefore contradicts all the phenomena 
and experiments which he supposed it to represent. But before we 
proceed to the proof of this assertion, it may perhaps be proper to 
remark, that previously to the date of M. Cauchy’s communication, 
and of my own paper, I had actually tried and rejected this identical 
hypothesis, and had even gone so far as to reject along with it the 
whole of M. Cauchy’s views about the mechanism of light. For 
though, in my paper, I have said nothing of any mechanical investi- 
gations, yet, as a matter of course, before it was read to the Academy, 
I made every effort to connect my equations in some way with me- 
chanical principles ; and it was because I had failed in doing so to my 
own satisfaction that I chose to publish the equations without com- 
ment* as bare geometrical assumptions, and contented myself with 
stating orally to the Academy, as I did some months after to the 
———————————— ee  <aT il Tie EEE ane 
cular and elliptic polarization in a lithographed memoir of the date of 
August 1836. But I do not find that he published, either then or since, 
the detailed calculations which he seems to have made. 
* The circumstances here related will account for what Mr. Whewel 
(History of the Inductive Sciences, vol. ii. p. 449) calls the “ obscure and 
oracular form” in which those equations were published. Having, at that 
time, no good explanation of them to give, I thought it better to attempt 
none, But in the general view which I have since taken, they do not offer 
any peculiar difficulty. 
