1. The following work is based directly on the mathematical treatment of the phenomena of after images given in Colour Vision, a recent book on the subject, by Professor Peddie.

The aim is to prove that this treatment agrees with the results obtained from experiments in which one eye was exposed to light for a given period. The oscillating decay of the resulting after images may be represented by the combined effect of three exponentially decaying periodic curves representing, respectively, the sensations of red, of blue, and of green.

In contributions to the theory of circulants this twenty-year period is found to have been still more fruitful than its fruitful predecessor, the number of writings belonging to it being, indeed, a fourth more, and the percentage of unoriginal communications much about the same. The language which appears oftenest is English, the next in order of frequency being Italian, French, German,—an unusual sequence: the American share of the first of these is considerable, and shows a tendency to increase. In our treatment of the subject papers on block circulants have not been kept separate from those on ordinary circulants: strict segregation has been found to necessitate considerable repetition, and the gain got from it in facility of reference is not so great as to overweigh this disadvantage.

In but a few cases are solutions of equations of the type

known, and even in the simplest cases the most general of the known solutions suffers from the disadvantage of being in a symbolic form. The equations to be studied in the present paper are those which can be derived from

by a change in the independent variable. This equation is the one which most naturally claims the attention of the investigator, after the equation of wave-motion

has been disposed of.