Page:Elementary Text-book of Physics (Anthony, 1897).djvu/426

412 centre $$l$$ fall on the mirror. Plainly they would be reflected to $$L$$ at a distance from the mirror less than $$\frac{r}{2},$$ as may be seen from the formula. If $$r$$ be negative, the centre is behind the mirror. The mirror is then convex, and the formula shows that for all positive values of $$p,$$ $$p'$$ is negative and numerically smaller than $$p.$$

337. Refraction at Spherical Surfaces.—The method of discussion which has been applied to reflection may be employed to study refraction at spherical surfaces. Let $$BD$$ (Fig. 103) be a spherical surface separating two transparent media. Let $$v$$ represent the velocity of light in the first medium, to the left, and $$v'$$ the velocity in the second medium, to the right, of $$BD.$$ Let $$L$$ be a radiant point, and $$mn$$ a surface representing the position which the wave surface would have occupied at a given instant had there been no change in the medium, $$m'n'$$ the wave surface as it exists at the same instant in the second medium in consequence of the different velocity of light in it, and $$l$$ the point where the prolongation of $$Bm'$$ backwards cuts the axis. We will investigate the conditions which must be fulfilled in order that the refracted wave shall appear to proceed from a point on the axis, or shall be a spherical wave.

In order that this should be the case, the time occupied by the light in travelling from the point $$l$$ on the axis with the velocity