546 
SECOND REPORT- 1832. 
thought desirable, by the Mathematical Committee of the Bri¬ 
tish Association for the Advancement of Science, that a short 
statement of this view of optics should be given in the forth¬ 
coming publication of that body. Such a statement, therefore, 
I shall now offer, as briefly as I can ; endeavouring only to 
communicate the view itself, and abstaining from giving any 
account of the results to which it has conducted me. 
“ The general problem that I have proposed to myself in 
optics, is to investigate the mathematical consequences of the 
law of least action : a general law of vision, in which are in¬ 
cluded, as it is well known, all the particular conditions of 
reflexion and refraction, gradual and sudden, ordinary and 
extraordinary. And the central idea from which my whole 
method flows, is the idea of one radical or characteristic relation 
for each optical system of rays , that is, for each combination of 
straight or bent, or curved paths, along which light is supposed 
to be propagated according to the law of least action. This 
characteristic relation, being different for different systems, and 
being such that the mathematical properties of the system can 
all be deduced from it, in the same manner as the method in¬ 
vented by Descartes for the algebraical solution of geometrical 
problems, flows all from the central idea of one radical relation, 
for each plane curve, or curved surface, in the form of which 
relation are included all the properties of the curve or the sur¬ 
face. In the radical relation thus contemplated by Descartes, 
in his view of algebraical geometry, the related things are ele¬ 
ments of position of a variable point which has for locus a curve 
or a surface ; and the number of these related elements is either 
two or three. In the relation contemplated by me, in my view 
of algebraical optics, the related things are, in general, in num¬ 
ber, eight: of which, six are elements of position of two vari¬ 
able points of space, considered as visually connected; the 
seventh is an index of colour; and the eighth, which I call the 
Characteristic Function, —because I find, that in the manner 
of its dependence on the seven foregoing are involved all the 
properties of the system,—is the action between the two vari¬ 
able points ; the word action being used here, in the same sense 
as in that known law of vision which has been already mentioned. 
I have assigned, for the variation of this characteristic function, 
corresponding to any infinitesimal variations in the positions on 
which it depends, a fundamental formula; and I consider as 
reducible to the study of this one characteristic f unction , by 
means of this one fundamental formula , all the problems of 
mathematical optics , respecting all imaginable combinations of 
mirrors, lenses, crystals and atmospheres. And though, among 
