Differential equation for normal glass dispersion and evaluation of the secondary spectrum

Some of the questions concerning secondary chromatic aberration at both sides of the visible band of the spectrum are the following: (1) What is the bandwidth at different wavelengths, given the permissible chromatic aberration circle and the lens aperture? (2) What is the size of the chromatic aberration circle, given the wavelength, the bandwidth, and the lens aperture? The answers to these and other questions may be found with the new definitions of V-number and relative partial dispersion P based on infinitesimal bandwidths that we propose. In addition, an alignment chart for the secondary color of a normal glass doublet is presented, so fast answers to the questions posed above and to other questions concerned with secondary color can be found. In addition, a continual challenge in computer-aided lens design is the use of optical glasses as design parameters in simultaneous optimization of lens systems over various regions of the spectrum. This problem could be solved if we could find an ideal glass family, not too different from real glasses, such that, given the refractive index n and the V-number at any wavelength, the indices at all wavelengths could be determined. Therefore we derive a differential equation for normal glass dispersion and present a recursive solution.