Page:Optics.djvu/118

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If we put $1+r$ and $1+v$ for the ratio of refraction belonging to the red and violet rays respectively, we shall have (taking for the general equation ${\frac {1}{F}}={\frac {m-1}{\rho }}$),

$Ar={\frac {\rho }{r}};\quad Av={\frac {\rho }{v}},$${\frac {Ar-Av}{Ar+Av}}={\frac {v-r}{v+r}}$,

and therefore if we put $a$ for the aperture $Bb,$

$no={\frac {v-r}{v+r}}\cdot a.$

Suppose, for instance, the lens be of crown-glass,

$v=.56$

$r=.54$,

${\frac {v-r}{v+r}}={\frac {.02}{1.1}}={\frac {1}{55}}.$

The diameter of the least circle of aberration is therefore ${\frac {1}{55}}$ of the aperture.

128. With regard to the distribution of the light over the surface of the circle of least aberration, it will be sufficient to observe that the vertices of all the cones of coloured light being on the axis of the lens, the center of the circle is one of them, so that it must be strongly illuminated, having the whole of the light of one sort thrown on it, besides portions of the others; the circumference on the contrary is enlightened only by the extreme rays of the red and blue cones, so that it is the least bright part, and it will be easily seen that the light diminishes gradually from the center of the circle to the edge.

On this account the effect of this aberration on images produced by lenses, is not so great as one might imagine from the great magnitude of the least circle of aberration: it certainly substitutes for