where ρ is the reciprocal of a radius and δn, or δn′, is the difference in refractive index between the rays chosen to be brought to exact focus together, as the red and the blue or violet.
This conventional equation simply states that the sum of the reciprocals of the radii of the crown lens multiplied by the dispersion of the crown, must equal the corresponding quantity for the flint lens if the two total dispersions are to annul each other, leaving the combination achromatic. Whatever glass is used the power of a lens made of it is
P( = 1/f) = Σρ(n - 1)
so that it will be seen that, other things being equal, a glass of high index of refraction tends to give moderate curves in an objective. Also, referring to the condition of achromatism, the greater the difference in dispersion between the two glasses the less curvatures will be required for a given focal length, a condition advantageous for various reasons.
The determination of achromatism for any pair of glasses and focal length is greatly facilitated by employing the auxiliary quantity ν which is tabulated in all lists of optical glass as a short cut to a somewhat less manageable algebraic expression. Using this we can figure achromatism for unity focal length at once,
P = ν/(ν-ν′)P′ = ν′/(ν-ν′)ν = (nD-1)/δn
being the powers of the leading and following lenses respectively. The combined lens will bring the rays of the two chosen colors, as red and blue, to focus at the same point on the axis. It does not necessarily give to the red and blue images of an object the same exact size. Failure in this respect is known as chromatic difference of magnification, but the fault is small and may generally be neglected in telescope objectives.
We have now seen how an objective may be made achromatic and of determinate focal length, but the solution is in terms of the sums of the respective curvatures of the crown and flint lenses, and gives no information about the radii of the individual surfaces. The relation between these is all-important in the final performance.
