ON DOUBLE REFRACTION. 271 
and in four positions the plane will touch the surface in one point, as repre- 
sented-in figs. 1, 2,4, 5. Should the contacts represented in figs. 4 & 5 take 
place simultaneously, they may be rendered successive by slightly altering 
the inclination of the plane. Hence in certain directions there would be four 
possible wave-velocities. Now the general principle of the superposition of 
small motions makes the laws of double refraction depend on those of the 
propagation of plane waves. But all theories respecting the propagation of a 
“series of plane waves haying a given direction, and in which the disturbance 
of the particles is arbitrary, but the same all over the front of a wave, agree 
in this, that they lead us to decompose the disturbance into three disturbances 
in three particular directions, to each of which corresponds a series of plare 
waves which are propagated with a determinate velocity. If the medium be 
incompressible, one of the wave-velocities becomes infinite, and one sheet of 
the wave surface moves off to infinity. The most general disturbance, 
subject to the condition of incompressibility, which requires that there be no 
‘displacements perpendicular to the fronts of the waves, may now be expressed 
as the resultant of two disturbances, corresponding to displacements in parti- 
cular directions lying in planes parallel to that of the waves, to each of 
which corresponds a determinate velocity of propagation. We see, therefore, 
that the limitation of the number of tangent planes to the wave-surface, 
which can be drawn in a given direction on one side of the centre, to two, or 
at the most three, is intimately bound up with the number of dimensions of 
space; so that the existence of the phenomenon of internal conical refraction 
is no proof of the truth of the particular form of wave-surface assigned by 
Fresnel rather than that to which some other theory would conduct. Were 
the law of wave-velocity expressed, for example, by the construction already 
mentioned having reference to the ellipsoid (12), the wave-surface (in this 
ease a surface of the 16th degree) would still have plane curves of contact 
with the tangent plane, which in this case also, as in the wave-surface of 
Fresnel, are, as I find, circles, though that they should be circles could not 
have been foreseen. 
_ The existence of external conical refraction depends upon the existence of 
a conical point in the wave-surface, by which the interior sheet passes to the 
exterior. The existence of a conical point is not, like that of a plane curve of 
contact, a necessary property of a wave-surface. Still it will readily be con- 
ceived that if Fresnel’s wave-surface be, as it undoubtedly is, at least a near 
approximation to the true wave-surface, and if the latter have, moreover, 
plane curves of contact with the tangent plane, the mode by which the 
exterior sheet passes within one of these plane curves into the interior will 
be very approximately by a conical point; so that in the impossibility of 
operating experimentally on mere rays the phenomena will not be sensibly 
different from what they would have been had the transition been made 
rigorously by a conical point. 
There is one direction within a biaxal crystal marked by a visible 
phenomenon of such a nature as to permit of observing the direction with 
precision, while it can also be calculated, on any particular theory of double 
refraction, in terms of the principal indices of refraction; I refer to the 
direction of either optic axis. Rudberg himself measured the inclination of 
the optic axes of Arragonite, probably with a piece of the same crystal 
from which his prisms were cut, and found it a little more than 32° as 
observed in air, but he speaks of the difficulty of measuring the angle with 
precision. The inclination within the crystal thence deduced is really a little 
greater than that given by Fresnel’s theory ; but in making the comparison 
