32 Mr. Mac Cutiacu on the dynamical Theory of 
Now let the figure TOSM be equal in all respects to QOPR, but in a posi- 
tion perpendicular to it, so that if QOPR were turned round in its own plane 
through a right angle, the point O being fixed, the points Q, P, R would fall 
upon T, S, M respectively ; and supposing the wave-plane ROy/’ to take various 
positions passing through O, imagine a construction similar to the preceding one 
to be always made by means of the two ellipsoids. Then while the points R and 
Q describe the ellipsoids, the points M and T describe two biaxal* surfaces 
reciprocal to each other, the latter surface being touched} in the point T by a 
plane which cuts OM perpendicularly in 8S. But this plane is parallel to the 
central wave-plane ROv/, and distant from it by an interval OS (= OP) which 
represents the velocity of the wave ; and as the surface whose tangent planes 
possess this property is, by definition, the wave-surface of the crystal, it is obvious 
that the point T describes the wave-surface. The radius OT, drawn to the 
point of contact, is then, by the theory of waves, the direction of the ray which 
belongs to the wave ROv’, and the length OT represents the velocity of light 
along the ray. As to the surface described by the point M, it is that which I 
have called the surface of indices, or the index-surface,§ because its radius OM, 
being the reciprocal of OS, represents the index of refraction, or the ratio of the 
sine of the angle of incidence to the sine of the angle of refraction, when the 
wave RQOy’, to which OM is perpendicular, is supposed to have passed into 
the crystal out of an ordinary medium in which the velocity of propagation is 
unity. The angles of incidence and refraction are understood to be the angles 
which the incident and refracted waves respectively make with the refracting 
surface of the crystal. 
The wave-surface and the index-surface have the same geometrical proper- 
ties, since they are both biaxal surfaces. Let us consider the former, which is 
generated by the ellipsoid whose semiaxes are a, b, c; and let us conceive this 
ellipsoid to be intersected by a concentric sphere of which the radius is 7. 
Then the equations of the ellipsoid and the sphere being respectively 
Sap 1, Gi Usha 
c r 
* See Transactions of the Royal Irish Academy, vol. xvii. p. 244. 
T Ibid. vol. xvi. p. 68. 
§ Ibid, vol. xviii. p. 38. I had previously called it the sufuce of refraction, vol. xvii. p. 252. 
