4.66 Mr. MacCutiacu on the Laws of 
It is commonly assumed that one of the rays is refracted according to the ordinary 
law; but this is not the case, since neither of the values of s is constant. However, 
the ray which has the greater velocity, (a being greater than b,) may still, for conve- 
nience, be called the ordinary ray. Of the two roots of equation (8), the one &,, 
whose numerical value (supposing @ not to vanish) is less than unity, corresponds to 
this ray. When C is positive, &, is negative ; and when C is negative, i, is positive : 
therefore in both kinds of quartz, by formuleze (5) and (6), we have s2> A, and s2<B; 
denoting by s, and s. the respective velocities of propagation of the ordinary and ex- 
traordinary waves. Hence, if we conceive a sphere of the radius a, with its centre 
at the origin, and a concentric prolate spheroid, whose semiaxis of revolution is also 
equal to a, and parallel to the axis of the crystal, while the radius of its equator is 
equal to b, the ordinary nappe of the wave surface will fall entirely without the sphere, 
_and the extraordinary nappe entirely within the spheroid, whether the crystal be right- 
handed or left-handed. With respect to the little ellipse in which the vibrations are 
performed, and of which the semiaxes parallel to x and y are represented by p and q 
respectively, it is evident that p>q for the ordinary wave, since k,<1 ; and that p<q 
for the extraordinary wave. When C vanishes, the minor axis of each ellipse also 
vanishes, and the rays become plane-polarized, the ordinary vibrations being then pa- 
rallel to the direction of x, and the extraordinary parallel to that of y. This is ex- 
actly what ought to happen on the supposition that the vibrations of a plane-polarized 
ray* are parallel to its plane of polarization ; a supposition which was kept in view 
in framing the fundamental equations (1.) and (2.). 
To show, with precision, how the two kinds of quartz are to be distinguished by 
the sign of C, we must give definite directions to the axes of coordinates. To this 
end, let us imagine the plane of xy to be horizontal, and a circle to be described in 
it with the origin O for its centre ; and let the north, east, and south points of this 
circle be marked respectively with the letters N, E, S. Let the direction of +z be 
eastward, from O to E; that of +y northward, from O to N; and that of +2 ver- 
tically downwards; the progress of the light through the crystal being also down- 
wards, and the plane of the wave moving parallel, as before, to the plane of zy. 
Then the crystal will be right handed or left handed, according as C is positive or 
* On this point there are two very different opinions. Fresnel supposed, as is well known, that the 
vibrations of a plane-polarized ray are perpendicular to its plane of polarization; whereas, according to M. 
Cauchy, whom I have followed, they are parallel to that plane. I am induced to adopt the latter sup- 
position, because I have succeeded, by means of hypotheses which are grounded on it, in discovering the 
laws of reflexion from crystallized surfaces ; laws which include, as a particular case, those discovered by 
Fresnel for ordinary media. The hypotheses alluded to, along with some of their results, are published 
in the London and Edinburgh Philosophical Magazine, vol. viii. p. 103, in a letter to Sir David Brewster. 
See also vol. vii. p. 295, of the same Journal. I hope soon to offer the Academy a detailed account of my 
researches on this subject. 
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