224 UNDULATORY THEORY OF LIGHT. 



•will be a series of sections, more and more inclined to QNQ', and also to QRQ', 

 and whose intersections with QNQ' will differ in azimuth along the arc HON, 

 which will have their axes parallel to LL' or P'P'. 



By considering the effect of turning the plane about CI, we should arrive at 

 a similar conclusion in regard to a series of sections cutting the circular section 

 near the point I, one axis of each of which would be parallel to CK, or normal 

 to P'P'. The normals to the planes of all these sections are the directions of 

 Avave progress, or nearly the directions of ray progress ; and if, from a point 

 above AA, the analyzer, lines should be drawn parallel to all those normals, 

 they would indicate the directions in which (no double refraction of the inci- 

 dent polarized ray occurring in them) the several points of the axis of the dark 

 bands or brushes ought to appear. These directions being all more inclined to 

 SC than is the normal to the circular section, it is evident that the pole in this 

 case will be the point of nearest approach of the dark band to the centre of 

 the field of view. It is furthermore evident that the bands are curved. For if 

 they are not so, the normals must all lie in one plane. But they cannot lie in 

 one plane unless the sections to which they are normals have a common inter- 

 section — a condition which, from the law of their construction, cannot exist. The 

 plane QNQ' turns about an axis movable in azimuth, and the surface Avhich is the 

 locus of all the normals is necessarily curved. 



The foregoing illustration accounts for only one of the dark bands. The other 

 is produced in the same way, and depends on the other circular section which is 

 not drawn. The analytic investigation of these changes would be extremely 

 complicated. 



It will be seen that this mode of explanation applies itself to the case of one- 

 axed crystals with great facility. The surface of elasticity for such crystals 

 being" an ellipsoid of revolution, every section has one of its axes in the plane 

 which contains its normal, and also the axis of the ellipsoid. The loci of the 

 dark bands will, therefore, always i:iecessarily be planes normal to each other, 

 intersecting in the optic axis of the crystal. 



The direction of ray-propagation is that of the radius of the wave. When 

 the theoretic wave is spherical, the ray is normal to the surface, but not other- 

 wise. The velocity of wave progress is measured by the normal let fall from 

 the centre of the wave upon the v/ave front; and this in spherical waves is the 

 same as the velocity of ray progress ; but in waves not spherical, ray progress 

 may exceed wave progress. 



§ XII. WAVE SURFACE. 



In order to determine, a priori, the direction which a ray will take on enter- 

 ing a doubly refracting medium it is necessary to know what is the figure of the 

 wave surface. For crystals of one axis we have seen that this problem was 

 solv(.'d by Iluyghens; but the complete generalization of the theory was reserved 

 for Frcsnel. 



Covdd a molecular movement be produced, starting from a single point and pro- 

 pagated in all directions in a medium of variable elasticity of three axes, the 

 surface defining the limits of the tremor at any moment would be the wave 

 surface. The same form of surface (sensibly) would be defined by an infinite 

 number of planes tangent to a luminous sphere like the sun, moving outwardly 

 from the body in all directions with velocities such as the law of variable 

 elasticity requires, when their distance from the body becomes very great 

 compared with the diameter of the sphere itself. Proceeding upon this 



