Page:Optics.djvu/113

 $$m, \ m'$$, and cutting it in $$k, \ k'.$$ It will easily be seen that all the retracted rays must pass between $$k$$ and $$k',$$ or that $$kk'$$ is the diameter of what in Chap. ix. we called the least circle of aberration or diffusion.

118.It is of some consequence to know how the light will be diffused over the area of this circle. It is easy to see that it will be most dense in the center and circumference; for the circumference is on the caustic where there are many rays crossing in a small space, and there is a cone of rays represented by the lines $$lo, \ l'o,$$ having its apex in $$o$$, the center.

119. $$ABC$$ (Fig. 132.) represent a straight line or rod sunk in water up to $$B$$. The rays of light proceeding from any point $$C$$ in the water will be refracted so as to proceed apparently from a point $$C',$$ which, by referring to Chap. vii., we find is higher than $$C$$ by one-fourth of its depth, (supposing $$m$$ to be $3⁄4$ for water).

The consequence of this is, that the visible appearance of the rod is such as is represented by the broken line $$ABC',$$ the tangent of the apparent angle of inclination to the surface ($$C'BD$$) being three-fourths of that of the real angle $$CBD$$.

120.Let us now examine the images produced by curved refracting surfaces. It will be quite sufficient in a practical point of view to take the cases of the double-concave and convex-lenses.

In these we find that when a pencil of rays is incident either along the axis or in a direction little inclined to it, the focal distance is given by the equation

$$\frac {1}{\Delta'} = \frac{1}{F} + \frac{1}{\Delta}.$$

From which we may conclude, that if $$\Delta$$ be constant, $$\Delta'$$ will be so; that is, if the object be a portion of a spherical surface, (Fig. 133.) having for its centre that of the lens, the image will be a similar portion of a sphere, the radius of which will be $$\Delta'$$ as that of the object is $$\Delta$$.

121.It the object be a straight line, it will be found that the case is precisely analogous to that in Chap. vi., substituting the center