Page:Elementary Text-book of Physics (Anthony, 1897).djvu/425

§ 336] and this point is therefore called a focus. Since the light passes through $$l,$$ it is a real focus. If $$l$$ were the radiant point, it is clear that the reflected light would be concentrated at $$L.$$ These two points are therefore called conjugate foci. If we divide both sides of the equation $$\frac{r}{p} + \frac{r}{p'} = 2$$ by $$r,$$ we have which is the usual form of the equation used to express the relation between the distances of the conjugate foci from the mirror.

A discussion of this equation leads to some interesting results. Suppose $$p = \infty,$$ then $$p' = \tfrac{1}{2}r;$$ that is, when the radiant is at an infinite distance from the mirror, the focus is midway between the mirror and the centre. In this case the incident wave is normal to the principal asis, and the focus is called the principal focus. Suppose $$p = r; \,\, p' = r$$ also. When $$p = \tfrac{1}{2}r, \, \, p' = \infty.$$ When $$p < \frac{r}{2}, \,\, \frac{1}{p} > \frac{2}{r}$$ and $$\frac{1}{p'} = \frac{2}{r} - \frac{1}{p},$$ a negative quantity. To interpret this negative result it should be remembered that all the distances in the formulas were assumed positive when measured from the mirror toward the source of light. A negative result means that the distance must be measured in the opposite direction, or behind the mirror. Fig. 103 represents this case. It is evident that the reflected wave is convex toward the region it is approaching, and proceeds as though it had come from $$l.$$ $$l$$ is therefore a virtual focus. Either of the other quantities of the formula may have negative values, $$p$$ will be negative if waves approaching their