Summary: 1. Fresnel's hypothesis—that light consists in the vibratory motion of an incompressible elastic ether—being untenable, should be abandoned as an educational instrument.

2. The later hypothesis—that light consists in the vibratory motion of a compressible elastic ether, of which the elasticity (of volume and figure) is the same for all bodies and for all directions in the same body, and of which the effective density in bi-refractive media is dependent on the direction of the vibratory motion—satisfactorily accounts for most of the known optical laws: hence such terms as “axes of optical elasticity,” which relate to variation of elasticity, must be discontinued.

3. Even this more satisfactory hypothesis may only be an approximate mechanical analogy, and may eventually be found to be inconsistent with experiment in some of its optical results; hence it cannot be satisfactorily used as the basis of a correlation of optical characters for the student of crystals; in fact, though it appears to be fully established that electromagnetic waves and light-waves differ only in length, an electro-magnetic disturbance seems to be inexplicable as mere vibratory motion of an elastic body.

4. On the other hand, the accuracy of Huygens' construction is now so far confirmed by experiment that it doubtless expresses a Law of Nature.

5. This being the case, it is easily seen that the velocity and polarisation of each of the two rays transmissible in a given direction in a uniaxal crystal can be simply expressed by means of the spheroid alone :—

If R be a point on the spheroid, O the centre, RN the normal, NOr a line intersecting the normal perpendicularly, the point R corresponds to a ray transmissible in the direction NOr with velocity 1RN. and plane of polarisation perpendicular to RN.

6. Generalisation suggests that, in the case of erystals belonging to a lower type of general symmetry, there is a similar correspondence between each ray and a point on an ellipsoid.

7. Experiment confirms the rigorous accuracy of the generalisation.

8. The surface of reference, whether a sphere, spheroid or ellipsoid, may be conveniently denoted by the term optical indicatrix.

9. All the optical characters can be direetly deduced from the indieatrix itself, and reference to its polar reciprocal is for this purpose unnecessary : further, it is possible to develop the characters from the consideration of rays alone.

10. The front of a pencil of rays which have started simultaneously from a point is part of the ray-surface; in the limit, if the pencil is of small aperture and includes a given ray, the pencil-front is part of the plane which touches the ray-surface where the ray meets it : hence the pencilfront corresponding to the given ray may be briefly designated as the ray-front.

11. A plane passing through a ray and perpendicular to its plane of polarisation may be conveniently termed its transverse plane.

12. In such case, it follows that the normal to the ray-front corresponding to the ray Or lies in the transverse plane RNOr and is perpen. dicular to OR, while the velocity of normal propagation of the front is measured by 1OR.

13. The normal RN is the direction of vibration of the ray corresponding to the point R, if the most recent hypothesis as to the proporties of an elastic luminferous ether is true.

14. The so-called primary and secondary optic axes are not axes of symmetry, nor even constant lines, of the crystal: they may with precision be denoted respectively as the optic bi-normals and bi-radials; for they are directions in which the two normals drawn from the centre to tangent planes of the ray-surface having the same direction, or the two radii vectores of the ray-surface having the same direction, are respectively coincident with each other. A crystal may still be loosely termed biaxal, when it is merely desired to suggest that the interference-rings shown by a plate in convergent polarised light are rudely like those which might be expected to be seen if the crystal had two axes, each identical in character with the optic axis of a tetragonal or hexagonal crystal.

15. By help of simple assumptions, which naturally present themselves and are consistent with all known experimental results, Fresnel's equation of the ray-surface. may be deduced from the general principles of undulations, without regard to the physical character of the periodic change.