42 Mr. Mac Cuttracu on the dynamical Theory of 
represent the different rays ; and let the length of each ray, measured from O 
in the direction of propagation, be assumed proportional to the velocity with 
which the light it propagated along it. Through the extremity of each ray con- 
ceive its transversal to be drawn, and let the transversals so drawn have their 
moments taken, with respect to the point O, as if they represented forces applied 
to a rigid body. The length of the incident or reflected ray being considered as 
unity, the lengths of the refracted rays (as appears by the last Section) are r and 
r’ respectively. Hence, as each transversal is perpendicular to its ray, the moments 
of the incident and reflected transversals are proportional to 7, 7;, and the 
moments of the refracted transversals to 77, 77, respectively. The equations (31) 
therefore signify that when the moments are projected, either upon the plane of 
Yo %» Or upon the plane of «, z,, the total projected moments are the same for 
the two media ; or that, if the transversals themselves be projected on either of 
these planes, the moments of the projections of the incident and reflected trans- 
versals are together equal to the moments of the projections of the refracted 
transversals. 
But the second of the equations (31) has another signification. For if the 
transversals applied at the extremities of the refracted rays be projected on the 
plane of a, z,, which is the plane of incidence, and contains the axes of z, and z;, 
the projections will be perpendicular to these axes, since the transversals them- 
selves are perpendicular to them; and the distances of the projections from the 
point O will be proportional to s and s’, or, by the relations (29), to sin 7, and 
sin ?,; so that if @, and 6, be the angles which the transversals make with the 
plane of incidence, the moments of the projections will be represented by 
7,C0s 0, sin 7, and 7; cos 6; sin 7,. At the same time, if 0, and 6; be the angles which the 
incident and reflected transversals make with the plane of incidence, the moments 
of the corresponding projections of these transversals will evidently be repre- 
sented by 7, cos 0, sin 7, and — 7/ cos 6; sin?,; the latter quantity being taken with 
a negative sign, because the extremity of the reflected ray, where the transversal 
7, is applied, lies in the first medium, while the extremities of the incident and 
refracted rays lie in the second, and it is supposed that when any of the angles 
O\ Oi, A, 05 1s zero, the direction of the corresponding transversal makes an acute 
angle with the axis of x, Hence we have 
7, COS 0, sin 7, — 7; Cos 6; sin 7, = 7, cos 0, sin Z, + 74 cos 65 sin 7, ; 
