44 Mr. Mac Cuttacu on the dynamical Theory of 
refracted transversal is the resultant of the other two, we have evidently 
, sin (2, — 2, Si sin 22, bay 
eee Cras ae D ee) 
2°. When the incident ray is polarized perpendicularly to the plane of inci- 
dence, the transversals are all perpendicular to that plane. Taking 2sin 7, to 
represent the length of the incident or reflected ray, the proportional length of 
the refracted ray is 2sin 2, and the projections of these lengths on the plane of 
Yo%, ave 2 sin z, cos 7, and 2 sin 7, cos 2,, or sin 22, and sin 27,. The transversals 
applied at the extremities of the rays are not altered by being projected on the 
plane of y,2z,; therefore the moments of the incident, reflected, and refracted 
transversals, projected on this plane, are represented by the quantities 7, sin 27,, 
— 7; sin 2z,, and 7, sin 2, respectively. Equating the last moment to the sum of 
the other two, and the refracted transversal to the sum of the other two trans- 
versals, we get 
(7, — 7;) sin 27, = 7, sin 22,, 1 
and thence 
tan (2, — 2,) sin 22 
‘ 1 2 1 ge 
= aay Sey ee 9 — = 7 ALLEN Ess Gan Len 33 
SO canes), 2 = 7 sin (4, + 2,) cos (¢, — 4) 7) 
This case has been considered by Fresnel. The relative magnitudes of the 
incident and reflected transversals, as given by him, are in accordance* with the 
formule (32) and (33); but with respect to the refracted transversals, his results 
do not agree with the formule. 
SECT. VI.—PRESERVATION OF VIS VIVA—THEOREM OF THE POLAR PLANE— 
CONCLUSION. 
Returning to the general question, if we resolve the transversals parallel to 
the axes of a, y, 2, and equate the sums of the parallel components in one 
* There is, however, a difference as to the relative directions of the incident and reflected trans- 
versals, When the second medium is the denser, and the incidence is perpendicular, these trans- 
versals, according to the present theory, have the same direction, but according to Fresnel they 
have opposite directions. 
