404 Royal Irish Academy: Prof. MacCullagh ona 
because it seemed probable that juster notions would prevail in the 
course of a few years, and that the ingenious speculations to which 
I have alluded would gradually come to be estimated at their proper 
value. But from whatever cause it has arisen—whether from the 
real difficulties of the subject, or the extreme vagueness of the ideas 
that most persons are content to form of it, or from deference to the 
authority of a distinguished mathematician—certain it is that the 
doctrines in question have not only been received without any ex- 
pression of dissent, but have been eagerly adopted, both in this 
country and abroad, by a host of followers; and even the extraor- 
dinary error, which it is my more immediate object to expose, has 
been continually gaining ground up to the very moment at which I 
write, and has at last begun to be ranked among the elementary 
truths of the undulatory theory of light. Notwithstanding my un- 
willingness, therefore, to be at all concerned in such discussions, I 
do not think myself at liberty to remain silent any longer. There 
are occasions on which every consideration of this kind must give 
way to a regard for the interests of science. 
To show that the principles of M. Cauchy contradict, instead of 
explaining, the phenomenon of elliptic polarization, let us take the 
axes of coordinates as before; and let us suppose, for the sake of 
simplicity, and to avoid his third ray, that the normal displacements 
vanish. Then his fundamental equations take the form 
a’é 
a wank HERS 
Raliat 
ae = UIAU+ ZhAE, 
where f, g, A are quantities depending on the law of force and the 
mutual distances of the molecules*. If, therefore, we assume that 
* T have not thought it necessary to transcribe the original equations of 
M. Cauchy, which are rather long. He has presented them in different 
forms ; but the system marked (16) at the end of § 1 of his Memoir on Di- 
spersion, already quoted, is the most convenient, and it is the one which I 
have here used. The directions of the coordinates being arbitrary, I have 
supposed the axis of z to be perpendicular to the wave-plane. Then, on 
putting € = 0, A f = 0, in order to get rid of the normal vibration, the 
last equation of the system becomes useless, and the other two are reduced ~ 
to the equations (2.) given above; the letters f,g,% being written in place 
of certain functions depending on the mutual actions of the molecules. It 
will be proved, further on, that this simplification does not at all affect the 
argument. As the directions of z and y still remain arbitrary, I have made 
them parallel to the axes of the supposed elliptic vibration. 
It may be right to observe, for the sake of clearness, that, when the me- 
dium is arranged symmetrically, it is always possible to take the directions 
of x and y such that the two sums depending on the quantity 4 may dis- 
appear from the equations (2.), and then the vibrations are rectilinear, But 
when the arrangement is unsymmetrical, this is no longer possible. 
The equations (2.) are precisely the same as those which have been em- 
ployed by Mr. Tovey and by Professor Powell, the latter of whom, in his 
lately published work, entitled * A General and Elementary View of the 
