48 Mr. Mac Cutracu on the dynamical Theory of 
both media, the equation expresses a principle analogous to that of the preserva- 
tion of vis viva.* 
By giving a certain direction to the incident transversal, that is, by polarising 
the incident ray in a certain plane, we may make one of the refracted rays dis- 
appear. If OT be the ray which remains, we have 7, = 0, and the equations 
(34) and (38) become 
(7, cos 0, + 7; cos 6;) Cos 7, = 7, COS A, COS 22, 
7, sin 0, + 7; sin 6; = 7, sin 65, 
(7, cos 6, — 7; cos 6) sin 7, = 7, COS 0, SIN 245 
(7, sin 6, — 7; sin 6) sin 2, cos 7, = 7, (sin @, sin 2, cos 7, + sin* 2, tan e). 
(43) 
In this case, the three transversals are in the same plane, the refracted transver- 
sal being the resultant of the other two. Therefore if we find this plane, every 
thing will be determined. 
The axes of x,, Y z, make with the incident transversal angles whose cosines 
are 
Cos 0, COs 2,5 sin 6, — cos 0, sin 2, 
and with the reflected transversal angles whose cosines are 
cos 6; cos 2,, sin 6), cos 6; sin 2, ; 
therefore, by Lemma I, the cosines of the angles which these axes make with a 
right line perpendicular to the plane of the transversals are proportional to the 
quantities 
sin (6, -+ 6) sin 2, — cos 6, cos 6; sin 22,, sin (6; — @,) cos %,. 
Now from the product of the first and second of the equations (43), combined 
with the product of the third and fourth, we find, by the help of the relations 
(37), 
27,7; sin (0, + 6;) sin 7, = 7,’tan 7, cos 6, {sin 0, cos?, —s*(sin @,cos?, +sin?, tan €)}. 
From the squares of the first and third of those equations we find 
— 2r, 7; cos 0, cos 6; sin 27, = 7,7 tan 7, cos? 6, (s? — 1), 
* A similar equation of vis viva holds when the light passes out of a crystal into an ordinary 
medium. 
