66 Professor Hamitton’s Zhird Supplement 
General geometrical Relations of infinitely near Rays. Classification of twenty- 
four independent Coefficients, which enter into the algebraical Expressions of these 
general Relations. Division of the general Discussion into four principal 
Problems. 
14. It is an important general problem of mathematical optics, included in that 
fundamental problem which was stated in the second number, to investigate the gene- 
ral relations of infinitely near rays, or paths of light ; and especially to examine how 
the extreme directions change, for any infinitely small changes of the extreme points, 
and of the colour: that is, in the notation of this Supplement, to examine the gene- 
ral dependence of the variations da, 9B, dy, da’, §8', dy’, on da, dy, Sz, da’, dy', dz’, dy. 
This important case of our fundamental problem is easily resolved by the application 
of our general methods, and by the partial differential coefficients, of the two first 
orders, of the characteristic and related functions: it may also be resolved by the 
partial differentials of the three first orders, of the characteristic function V alone. 
For from these we can in general deduce six linear expressions for da, 83, dy, da, 8B", 
dy’, in terms of dz, dy, dz, da", dy’, d2', dy, involving forty-two coefficients, of which 
however only twenty-four are independent, because they are connected by fourteen re- 
lations included in the formule asa + B88 + y8y=0, a'8a' + B'9B' + y'dy'=0, and by four 
more included in the conditions that the final direction does not change when the 
initial point takes any new position on the given luminous path, nor the initial direc- 
tion when the final point is removed to any new point on that given path. 
Thus, if we employ the characteristic function 7, and the final and initial medium- 
functions v, v', we have, by (B), the following general relations : 
OV ov OV bv OV bv 
ra Sat ae. Age ie aye A’) 
pat ote Nae Sh ae OS or aay 
ae) a a One Pes. 8y/' = yeu eae Sela 3y’ ~ 
in which, by the last number, we are at liberty to assign different origins and different 
and oblique directions to the axes of the final and initial co-ordinates, if we assign 
new and corresponding values to the marks of final and initial direction, a, [, y, 
a’, ', y', so as to have still the equations her 
==, oy oi dz! 
=a Say ae ds' mie a aa 
ds being still the final, and ds' the initial element of the curved or polygon path. We 
may suppose, for example, that both sets of co-ordinates are rectangular, but that the 
origins of the final and initial co-ordinates are respectively the final and initial points 
