of the image formed by a lens, in the cases of both central and eccentric oblique refractions. He assumed the aperture of the pencil of rays in question to be infinitely narrow, and got at his results by the employment of the differential calculus. I saw that while this would be quite valid for such infinitely narrow pencils, still, as considerably broad pencils generally occur in practice, it struck me it might be worth while trying to devise a method not dependent upon the calculus, whereby the foci of broad oblique and eccentric pencils could be elucidated, when possibly some new results of practical importance might be forthcoming. About the year 1890 I undertook that task, and after meeting with many difficulties which almost compelled me to give up the investigation as hopeless, I at last succeeded in arriving at the results embodied in Sections V., VI., and VII. of this volume, and in so doing was fortunate enough to bring to light the formula relating to coma, a phenomenon that appears, strange as the fact may seem, never to have been noticed by Coddington. I then saw that the formulæ I thus arrived at implied corollaries of the greatest practical importance, and I was led almost directly to the conception of the Cooke lens, that is, of the older complex Cooke lens built up of two achromatic positive lenses and one achromatic negative lens. The simple Cooke lens was of later conception. Thus the theory preceded the practice, although I should say that there are certain other features of the Cooke lens, such as distortion and oblique achromatism more especially, whose theory I did not arrive at until a few years later, so that in that respect the practice preceded the theory.
Having subsequently worked out a complete system of formulæ, which I have proved and tested and found reliable in all manner of ways, and recognising the great importance of theory and practice working loyally together for future improvements, I thought that as soon as I had time enough at my disposal I would gather together and arrange what has been the interrupted labour of many years, with a view to publication, if by so doing I could, even in a humble degree, forward the development of optical science in this country, wherein it has lain so long neglected, or perhaps furnish some raw material on which some far abler heads than mine should at some future time found important corollaries not yet dreamed of.
Considerations of space have compelled me to confine myself to theorems and formulæ that I consider to be of the greatest practical value, and to leave out many corollaries of minor importance that might be dealt with in a future edition, were it ever called for.
There are also many problems and theorems untouched upon,
