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single foci, so many interconfused circles, but as these are bright only at their centers, and above all, as the yellow light, which is the brightest in the spectrum, converges nearly to those centers, the haziness is not very considerable except in cases where the light is very much condensed by a lens of short focal length.

129.The chromatic aberration is a much more serious bar to the perfection of optical instruments depending on the lens, than that owing to the spherical figure, for this latter imperfection can be made quite insensible in most cases, by diminishing the aperture of the lens, since it varies as the square of this line, whereas the former varying as the simple power of the aperture, will be diminished certainly, but very considerably less than the other.

It has therefore been a great desideratum to find some way of constructing a lens, so as to be achromatic, and this has been tolerably well effected, by joining together two or more lenses, made of substances having different dispersive powers, so that the dispersions may be equal and opposite, though the refraction be not wholly destroyed.

130.The expression for the principal focal length^[1] of a combination of lenses placed close together was found (p. 68.) to be

${\frac {1}{\phi }}={\frac {m-1}{\rho }}+{\frac {m'-1}{\rho '}}+{\frac {m''-1}{\rho ''}}+....$

If therefore $1+r$ and $1+v$ represent the values of $m$ for red and violet rays, we shall have, taking only two lenses,

${\frac {1}{\phi }}={\frac {r}{\rho }}+{\frac {r'}{\rho '}},$ for the red ray,

${\frac {1}{\phi }}={\frac {v}{\rho }}+{\frac {v'}{\rho '}},$ for the violet.

Now it is clear that if we chuse to leave $\rho$ and $\rho '$ indeterminate, we may equate these two values of $\phi ,$ and so obtain proper values for $\rho$ and $\rho ',$

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1. It is quite sufficient to consider the principal focal length, as it will easily be seen, that in the case of rays not parallel, one has only to add ${\frac {1}{\Delta }}$ to the expression.