102 Professor Hamitton’s Third Supplement 
involving only the extreme points, directions, and colour, of the given luminous path, 
and the properties of the extreme media. 
The simplest manner of obtaining these ten general relations, is to eliminate the 
fourteen differentials of 7 which enter into the twenty-four expressions, deducible 
from (C°), from the twenty-four coefficients (D"). The ten relations thus obtained, 
may be arranged in three different groupes: the first groupe containing the two 
following 
A® 
cv da ov 6B Sv _ wv Sr 
SadpB Sz 1 9p? dz | 8882 — 
and two others similar to these, but with accented or initial symbols ; the second 
groupe containing the final relation 
Sv da dv eB ev sv oa Sv OB ov (B") 
Sa2 Sy + Sad8 dy | Sady  dadp Oe Se 8x” §BSx’ 
and a similar initial relation ; and the third groupe comprising the four following, 
Sv da Sv OB oul da yc 6p 
Sa Sc’ * Sad de’ | Sa? Ox "eB ae 
Sv Ba, 8 8B Sal Se BB _ 
Sa? By’ | SaB dy * Sa'dp’ Ox | S82 Ba is 
Bo Ba, 7 8B, Bi BY BW 8B cc) 
Sad Sx" | 32 Oe | Sa? Sy | Sade’ dy 
cv 6a nee ep set Sv! oa’ mi Sv’ op’ = 
3adB dy’ * 3G" dy’ | dads’ dy | SB dy 
The two first relations of the first groupe, namely, the equations (4), are equi- 
valent to the two first differential equations (0) of a curved ray, and express that the 
magnitude and plane of final curvature of a luminous path, in a final variable medium, 
are determined, in general, by the properties of that medium, the colour of the light, 
the position of the final point, and the direction of the final tangent. And the two 
other relations of the same groupe express, in like manner, a dependence of the 
initial magnitude and plane of curvature of a luminous path, on the initial medium, 
colour, point, and tangent. 
The equation (B"), belonging to the second groupe, is a relation between the four 
coefficients 7 ; = e = , and therefore a relation between the guiding paraboloid 
and constant of deviation Pi the final ray-lines, depending on the final medium, 
colour, point, and tangent. And similarly the other equation of the second sroune 
expresses an analogous relation for the initial medium. 
