Page:Elementary Text-book of Physics (Anthony, 1897).djvu/427

§ 337] which it has in the second medium should be the same for all points on the refracted wave. This time is given by $$t' = \frac{lm'}{v'} = \frac{lB + Bm'}{v'};$$ we are to investigate the conditions that this shall be the same for all points on the refracted wave.

The time occupied by the light in travelling from $$L$$ to $$m'$$ is $$\frac{LB}{v} + \frac{Bm'}{v'} = C,$$ a constant for all points on the refracted wave. Subtracting from this the expression for $$t'$$ we have $$C - t' = \frac{LB}{v} - \frac{lB}{v'},$$ and the condition that $$t'$$ should be constant is therefore that $$\frac{LB}{v} - \frac{lB}{v'}$$ is constant. Since $$\frac{v}{v'} = \mu,$$ we may write the expression which should be constant $$LB - \mu lB = LA - \mu lA.$$

Using the notation of the last section, and substituting the values of $$LB$$ and $$lB$$ as there found, except that $$p$$ is used instead of $$p',$$ we have as the condition that $$t'$$ is a constant, whatever be the value of $$x$$ within the limits set to it, the equation From this we obtain $$\frac{r}{p} - \frac{\mu r}{p} = 1 - \mu,$$ and  Hence the point at the distance $$p''$$ from the centre of the refracting surface is the centre of a spherical refracted wave.

If the medium to the right of $$BD$$ be bounded by a second spherical surface, it constitutes a lens. Suppose this second surface to be concave toward $$l$$ and to have its centre on $$AC.$$ The wave $$m'n',$$ in passing out at this second surface, suffers a new change of form precisely analogous to that occurring at the first surface, and the new centre is given by the formula just deduced by substituting for $$p$$ the distance of the wave centre from the new surface, and for $$\mu$$ the index of refraction of the third medium in relation to the second. If $$s$$ represent the distance of $$l$$ from the new