400 Royal Irish Academy: Prof. MacCullagh on a 
roche) n’est pas rigoureusement circulaire, mais qu’alors l’ellipse dif- 
fére trés peu du cercle.” : 
Thus, to say nothing for the present of the questions of disper- 
sion and absorption, it appears that M. Cauchy conceived he had 
completely accounted for the facts of circular and elliptic polariza- 
tion, and that he had deduced the formulas ‘‘ which serve to deter- 
mine the conditions and laws of these phenomena.” But neither 
in this letter, nor in any subsequent version* of his theory, has he 
given the formulas themselves. Nor has ‘he told us the nature of 
the calculations by which he was enabled to correct the received 
opinion, and to prove that the vibrations in a ray transmitted along 
the axis of quartz, or through oil of turpentine, are not rigorously 
circular as Fresnel and others have supposed, but slightly elliptical. 
Now—to take the case of quartz—if we consider that the vibrations 
of a ray passing along the axis are in a plane perpendicular to it, and 
if we admit, as M. Cauchy always does in the case of other uniaxal 
crystals, that there is a perfect optical symmetry all round the axis, 
we shall find it hard to conceive on what grounds he could have 
come to the conclusion that the vibrations of such a ray are performed 
in an ellipse. For if all planes passing through the axis of the 
crystal be alike in their optical properties, there will be absolutely 
nothing to determine the position and ratio of the axes of the ellipse ; 
there will be no reason why its major axis, for example, should lie 
in one of these planes rather than in any other. But, whatever 
may be thought of this case independently of observation, it is mani- 
festly absurd to suppose that the vibrations are elliptical in the 
case of a ray passing through oil of turpentine, or any other liquid 
possessing the property of rotatory polarization ; for, in a liquid, all 
planes drawn through the ray itself are circumstanced alike. From 
these simple considerations it is evident that the theory of M. Cauchy 
is unsound; but a closer examination will show that it is entirely 
without foundation, and that it is directly opposed to the very phe- 
nomena which it professes to explain. ‘To make this appear, how- 
ever, in the easiest way that the abstruseness of the subject will allow, 
it will be necessary to advert to some former researches of my own, 
which have a direct bearing on the question. 
The same day on which M. Cauchy’s letter was read to the French 
Academy, I had the honour of reading to the Royal Irish Academy a 
paper ‘‘ On the Laws of Double Refraction in Quartz” (see Transac- 
tions of the Royal Irish Academy, vol. xvil. p. 461), wherein I showed 
that everything which we know respecting the action of that crystal 
upon light is comprised mathematically in the following equations :— 
dee UGE My 
ae Ge to ae i) 
d2 4 O41 _¢ BE 
d t? dz? dz’ 
* From some statements that have been made within the last few days 
by Professor Powell (Phil. Mag., S. 3.vol. xix. p. 374), at the request of M. 
Cauchy himself, it appears that the latter republished his views about cir- 
