2 Professor Hamitton’s Third Supplement 
of the final and initial co-ordinates, that is, the variation of the action, or the time, 
expended by light of any one colour, in going from one variable point to another, is 
sV=(sfvds = eae = ba" + a = oy’ + = be Be (A) 
the accented being the initial quantities. This general equation, (4), which I have 
called the Equation of the Characteristic Function, involves very various and exten- 
sive consequences, and appears to me to include the whole of mathematical optics. I 
propose, in the present Supplement, to offer some additional remarks and methods, 
connected with the characteristic function 7, and the fundamental formula (A) ; and 
in particular to point out a new view of the auxiliary function 7’, introduced in my 
former memoirs, and a new auxiliary function 7, which may be employed with advan- 
tage in many optical researches : I shall also give some other general transformations 
and applications of the fundamental formula, and shall speak of the connection of my 
view of optics with the undulatory theory of light. 
Fundamental Problem of Mathematical Optics, and Solution by the Fundamental 
Formula. Partial Differential Equations, respecting the Characteristic Function 
FV’, and common to all optical combinations. Deduction of the Medium Functions 
Q, v, from this Characteristic Function V. Remarks on the new symbols o, 7, v. 
2. It may be considered as a fundamental problem in Mathematical Optics, to 
which all others are reducible, to determine, for any proposed combination of media, 
the law of dependence of the two extreme directions of a curved or polygon ray, 
ordinary or extraordinary, on the positions of the two extreme points which are 
visually connected by that ray, and on the colour of the light: that is, m our present 
notation, to determine the law of dependence of the extreme direction-cosines a 3 y 
a By, on the extreme co-ordinates x y z v' y' 2, and on the chromatic dex yx. 
This fundamental problem is resolved by our fundamental formula (4); or by the 
six following equations into which (4) resolves itself, and which express the law of 
dependence required : 
SV fv OVA See OF _ ob 
3c ~ Sa? y 7 5B? de — By? 
SV Bey A eee. meta 
—3,/= 5,9 — 3y' = 3p" ; Tay ate) ¢ 
These equations appear to require, for their application to any proposed combination, 
not only the knowledge of the form of the Characteristic Function V’, that is, 
the law of dependence of the action or time on the extreme positions and on the 
colour, but also the knowledge of the forms of the functions v, v’, that is, the optical 
