72 Professor Hamitton’s Third Supplement 
and in all these equations of curvature we may, consistently with hes ction of the 
e F a because 
they relate to motion along a given luminous path It is evident that these coeff- 
: ca 0 
present Supplement, express the coefficients = ; =e by the symbols “2 
cients vanish, when the final portion of this path is straight. But when this final 
portion is curved, we may geometrically construct the effect of the curvature on the 
final direction, by making the final element ds revolve through an infinitely small 
angle round the final axis of curvature. 
The two remaining problems are more complicated, because each involves two 
independent variations dr, ey, namely the two rectangular co-ordinates of the near 
point B, on the final plane of 2y, which point is considered as the final point of a 
near luminous path. The equations of the right line, which is the final portion or 
final tangent of this near path, are, 
G=00 + 2(= ou ie ay ) ) 
ox oy 
Ge) 
y=yteZ ie dict +Py)s 
and the equations of the right line which is the initial portion or the initial tangent 
of the same near path, are 
Be (a+ ay ) 5 au 
y= 2 (Lars 2 x) il 
a Tk 
Our third problem is to investigate the geometrical relations of the system of right 
lines (V”"), which we shall call final ray-lines, with each other, and with the co-ordi- 
nates 6x, dy; and our fourth problem is to investigate the connexion of the same 
co-ordinates or variations with the right lines of the system (//”*), which may be 
called initial ray-lines. 
The third problem may be considered as resolved, if we can assign any surface to 
which the final ray-lines (/”°) are normals, or with which they are determinately con- 
nected by any other known geometrical relation. Let us therefore examine whether 
the ray-lines of the system (/”*) are normals to any common surface, which passes 
through the given final point of the original luminous path. If so, this surface may be 
considered, in our present order of approximation, as perpendicular to the final rays 
themselves. Now, in general, when rays of a given colour diverge from a given lumi- 
nous point, and undergo any number of ordinary or extraordinary and gradual or 
sudden reflexions or refractions, they are, or are not, perpendicular in their final state 
to a common surface, according as the following differential equation 
ad + Boy +82 = (X°) 
