Page:Optics.djvu/120

 $$\frac{r}{\rho}+\frac{r'}{\rho'}=\frac{v}{\rho}+\frac{v'}{\rho'},$$

$$r\rho'+ r'\rho=v\rho'+v'\rho, \mathrm{or} \ \frac{\rho'}{\rho} = -\frac{v'-r'}{v-r},$$

which shows that $$\rho'$$ and $$\rho$$ must be of different signs, or one lens concave and the other convex; and that they are as the respective dispersions of the lenses.

In order therefore to bring the most unequally refrangible rays, namely, the red and violet, to one focus, we have only to put together a convex and a concave-lens, and to make the quantities represented by $$\rho$$ and $$\rho'$$ (or the principal focal lengths, for any one kind of simple light, which are in the same ratio) proportional to the dispersive powers of the substances, of which the lenses are made.

131. The common practice of opticians is to use flint-glass, and crown-glass, the dispersive powers of which are in the ratio of 50 to 33; and therefore a compound-lens such as that represented in Fig. 141. in which the separate focal lengths, for the same kind of homogeneous light, are as 50: 33, will make the red and violet rays of the solar, or any similar light, converge accurately to one point.

To illustrate this, let $$v, g, r,$$ (Fig. 142.) be the points to which the convex-lens by itself would throw the violet, green, and red rays. The addition of a concave-lens diminishes the convergency, and therefore throws the foci farther off; but it affects the violet and green light more than the red, so that they are all brought closer together, and if the lenses be so matched that the dispersive power of the first is just balanced in all parts by the counter-dispersive power of the second, the rays will all be brought to one single point $$f.$$

If, however, the substances of which the two lenses are made, do not act with equal inequality, on the different coloured rays, the object will not be attained. If for instance, (Fig. 143.) the first lens disperses the rays so that the foci $$v, g, r$$ are equidistant, but the second lens acts very nearly as strongly on the green rays as on the violet, it may throw the red focus from $$r$$ to $$r',$$ the violet from $$v$$ to $$v'$$ close to it, $$vv'$$ being greater than $$rr',$$ but the green will go from $$g$$ to $$g'$$ making $$gg'$$ nearly equal to $$vv',$$ so that the three foci will not coincide.