ON DOUBLE REFRACTION. 273 
emergence (according as the prism remains fixed or turns round with the tele- 
scope) be measured, by observing the light reflected from the surface, and like- 
wise the deviation for several standard fixed lines in the spectrum of each 
refracted pencil. Let the prism be now turned into a different azimuth, and the 
deviations again observed, and so on. Each observation furnishes accurately 
an angle of incidence and the corresponding angle of emergence; for if ¢ be 
the angle of incidence, 7 the angle of the prism, D the deviation, and y the 
angle of emergence, D=¢+y—i. But without making any supposition as 
to the law of double refraction, or assuming anything beyond the truth of 
Huygens’s principle, which, following directly from the general principle of 
the superposition of small motions, lies at the very foundation of the whole 
theory of undulations, we may at once deduce from the angles of incidence 
and emergence the direction and velocity of propagation of the wave within 
the prism. For if a plane wave be incident on a plane surface bounding a 
medium of any kind, either ordinary or doubly refracting, it follows directly 
from Huygens’s principle that the refracted wave or waves will be plane, and 
that if g be the angle of incidence, g! the inclination of a refracted wave to 
the surface, V the velocity of propagation in air, v the wave-velocity within 
the medium, 
sin ¢_ sin q! 
4 boknienas ae 
Hence if g', )! be the inclinations of the refracted wave to the faces of our 
prism, we shall have the equations 
0 Ba 6 = Voegeli eis ied wl, oe 218) 
Pi Wee, Toe ee eee iy Ord (14) 
Sher dane Get eee en 
The equations (13) and (14) give, on taking account of (15), 
: — ee ‘J 
v sin SFY os PSY RV sin 5 cos HEY hii meray. 
pn CO argh es Cone. 
v cos 5+ aint =V cos 5 ain ao » « (17) 
whence by division 
da Oe a gM Gu 
tan —z = tan 5 tan = cot as ig haa (18) 
The equations (15) and (18) determine g’ and y', and then (16) gives v. 
Hence we know accurately the velocity of propagation of a wave, the normal 
to which lies in a plane perpendicular to the faces of the prism, and makes 
known angles with the faces, and is therefore known in direction with 
reference to the crystallographic axes. A single prism would enable the 
observer to explore the crystal in a series of directions lying in a plane 
perpendicular to its edge; but as these directions are practically confined 
to limits making no very great angles with a normal to the plane bisecting 
the dihedral angle of the prism, more than one prism would be required to 
enable him to explore the crystal in the most important directions; and it 
would be necessary for him to assure himself that the specimens of crystal, 
of which the different prisms are made, were strictly comparable with each 
hac It would be best, as far as practicable, to cut them from the same 
ock. 
The existence of principal planes, or planes of optical symmetry, for light 
r 
