406 Royal Irish Academy: Prof. MacCullagh on a 
Qr if 297 C 
f= A ———OCk, aM lahat ar at ae (6.) 
and thence 
r 
ii oe — — — 
k aac (A BB) 0) re > sien cproustbaden ste GAs) 
a result which is perfectly inconsistent with the former, since the 
two roots of (5.) have the same sign, if they are not imaginary, while 
those of (7.) have opposite signs, and cannot be imaginary. If, there- 
fore, one equation agrees with the phenomena, the other must con- 
tradict them. The last equation indicates that, in the double re- 
fraction of quartz, the two elliptic vibrations are always possible, 
and performed in opposite directions, which is in accordance with 
the facts; whereas the equation (5.), deduced from M. Cauchy’s 
theory, would inform us that the vibrations of the two rays are either 
impossible or in the same direction*, 
To apply the results to a particular instance, let us conceive 
a circularly polarized ray passing along the axis of quartz, or 
through one of the rotatory liquids, such as oil of turpentine; the 
position of the coordinates w and y, in the plane of the wave, being 
now, of course, arbitrary. In each of these cases we have k = + 1, 
and A = B= a?, so that the value of s® in equation (6.) is expressed 
by the constant a®, plus or minus a term which is inversely propor- 
tional to the wave-length A; the sign of this term depending on the 
direction of the circular vibration. Now it will not be possible to 
obtain a similarvalue of s? from the formulas (4.), unless we suppose 
A! = B! = a?, since it is only in the expansion of C’ that a term in- 
versely proportional to A can be found; but on this supposition the 
formulas are inconsistent with each other, nor can they be reconciled 
by any value of k. Indeed, when A' = B’, the equation (5.) gives 
k = + “%—1. Thus it appears that circular vibrations, such as are 
known to be propagated along the axis of quartz, and through cer- 
tain fluids, cannot possibly exist on the hypothesis of M. Cauchy. 
It was probably some partial perception of this fact that caused M. 
Cauchy to assert that the vibrations in these cases are not exactly 
circular, but in some degree elliptical; a supposition, which, if it 
were at all conceivable, which we have seen it is not (p. 400), would 
be at once set aside by what has just been proved; for no assumed 
value of &, whether small or great, will in any way help to remove 
the difficulty. 
But this is not all, Rectilinear vibrations are excluded as well 
as circular; for we cannot suppose k = 0 in the equations (4.), so 
long as the quantity C’, resulting from the hypothesis of unsymme- 
trical arrangement, has any existence. Thus the inconsistency of 
that hypothesis is complete, and the equations to which it leads are 
utterly devoid of meaning. 
* This conclusion, which shows that M, Cauchy’s theory is in direct 
opposition to the phenomena, might have been obtained without any re- 
ference to the equations (1.).. But these equations are necessary in what 
follows, 
