THIRD SUPPLEMENT. 
Fundamental Formula of Mathematical Optics. Design of the present Supplement. 
1. Wuen light is considered as propagated, according to that known general law 
which is called the law of least action, or of swiftest propagation, along any curved 
or polygon ray, ordinary or extraordinary, describing each element of that ray 
ds= ¥ (da + dy’ + dz*) with a molecular velocity or undulatory slowness v, which is 
supposed to depend, in the most general case, on the nature of the medium, the posi- 
tion and direction of the element, and the colour of the light, having only a finite 
number of values when these are given, and being therefore a function of the three 
rectangular co-ordinates, or marks of position, 7, y, z, the three differential ratios or 
cosines of direction, 
di dz 
o= Bap Y= as? 
and a chromatic index or measure of colour, x, the form of which function v depends 
on and characterises the medium; then if we denote as follows the variation of this 
function, 
Bo se by +e Pae4 Pea BoB += ay +e dx, 
and if, by the help of the relation a* + 6° + y°=1, we determine 
By Bo Be 
Sa? yey 8 va 
so as to satisfy the condition 
namely, by making v homogeneous of the first dimension with respect to a, B, y; it 
has been shown, in my First Supplement, that the variation of the definite integral 
V =f vds, considered as a function, which I haye called the Characteristic Function 
B 
