481 
Monpay, Fepruary 23RD, 1857. 
JAMES HENTHORN TODD, D.D., Presipent, 
in the Chair. 
Rev. Georce Satmon read a Paper by Mr. A. Cayley, on 
Professor Mac Cullagh’s theorem of the polar plane. 
‘A ray of polarized light, incident on the surface of an 
extraordinary medium, may give rise to a reflected ray and a 
single refracted ray ; but this will be the case only for a par- 
ticular position, or positions, of the plane of polarization of the 
incident ray. According to Professor Mac Cullagh’s theory, 
the planes of polarization, and the relative vibrations of the 
three rays, are deduced from two assumed principles, which 
may be referred to as— 
‘1°. The principle of equivalent vibrations. 
‘62°. The principle of equivalent moments. 
And from these principles are deduced,— 
‘¢ 3°. The principle of vis viva. 
«4°, The theorem of the polar plane. 
‘¢ The directions of the vibrations are completely deter- 
mined by means of 4°, the theorem of the polar plane; and the 
relative magnitudes are then given by 1°, the principle of equi- 
valent vibrations. The other principles, viz. :—2°, the prin- 
ciples of equivalent moments, and 3°, the principle of vis viva, 
must therefore follow as mere geometrical consequences from 
the first-mentioned two principles, or theorems; and I have 
found that the deduction depends immediately upon the fol- 
lowing two theorems in spherical trigonometry. 
‘¢ Suppose (Fig. 1, on next page) that RR'R” isa spheri- 
cal triangle, and let W be any point in the base RR”, and NV 
be the central point of the base; then joining WR’ and pro- 
2x2 
