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The first term of which differs but little from that found before, as $AT$ or ${\frac {\operatorname {ver~sin} \theta }{\cos \theta }}$ differs but little from $\operatorname {ver~sin} \theta$ when $\cos \theta$ is so nearly equal to $1.$

When the incident rays are parallel, the aberration is ${\frac {f}{\cos \theta }}-f$ or 1/2$AT$ accurately.

If $\theta =30^{\circ }$, $\alpha =\left({\frac {2}{\sqrt {3}}}-1\right)f;$ for $\theta =45,$ $\alpha =({\sqrt {2}}-1)f;$ for $\theta =60,$ $\alpha =f;$ so that the point $v$ coincides with $A.$

 

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CHAP. IV.

REFLEXION AT CURVED SURFACES NOT SPHERICAL.

17.⁠It will be recollected that the common parabola has the peculiar property that two lines drawn from any point on the curve, one to the focus, the other parallel to the axis, make equal angles with the tangent or normal; whence it follows that rays proceeding from the focus of a paraboloid will be reflected accurately parallel to its axis, and vice versâ rays coming parallel to the axis will be made to converge accurately to the focus or from it according as it is the concave or convex surface that reflects.

18.⁠In like manner rays proceeding from one focus of an ellipsoid will be reflected accurately to the other focus, or if the outward be the reflecting surface, rays converging towards one focus, will diverge after reflection as if they proceeded from the other.

19.⁠The analogous property of the hyperbola leads one to the conclusion that the surface generated by the revolution of that figure about its major axis, is such that rays meeting in one focus, will after reflexion diverge from or converge to the other.