218 
NOTES AND MEMOEANDA. 
determined by reference to the edges of the cube if the three plane 
angles are known which the three projections of the face of the crystal 
make with the three edges of the cube. Two plane angles even are 
sufficient, for the third can be calculated from the first two by the 
simple formula 
tan. a — cot. h cot. c ; 
a, 5, c being the plane angles which the three projections of the face of 
the crystal make with three edges of the cube meeting at the same 
summit. 
A second face of the crystal will be equally determined as regards 
its direction, by the three angles a, y, these three angles corre- 
sponding to the angles a, Z>, c, of the first face of the crystal, as was said 
above. 
It follows therefore that if the three angles a, 6, c, are known, or 
two only of those angles, and the three angles a, y, or two only of 
those angles, we can calculate the dihedral angle of the two faces of 
the crystal by the formulae 
cos. X 
cos. y sin. (z — (p) 
sin. (p 
cot. (p = tan. y cos. (6 + j8), 
tan. y 
tan. a 
tan. ^ = 
tan. a 
cos. b cos. iS 
If X is indeterminate, it can be calculated by the formulae 
sin. J X 
cos. ^ (y + z) 
tan. 
sin, i (b -[- $) 
cos. H*/ + ^) 
sin. y sin. 
It only remains then to point out a practical means of measuring 
the angles a, h, c, a, y6, y. 
I place in the eye-piece of a microscope a cylinder of flint glass 
whose index of refraction is greater than that of Canada balsam. This 
cylinder (the two bases of which are exactly parallel) is divided in 
halves by a plane perpendicular to the bases; the two rectangular 
faces are polished and fixed together again by Canada balsam. This 
cylinder is placed in the eye-piece so that its upper base is in the 
focus of the upper lens of the eye-piece, the two bases being perpen- 
dicular to the optic axis of the microscope, and the median plane of 
the cylinder passing through the optic axis and through the zero of the 
division of the revolving stage. 
Under these conditions if the microscope receives the light in a 
direction parallel to the median plane of the cylinder, an illuminated 
field will be seen crossed by a line forming a reticle ; but if the micro- 
scope receives the light obliquely to the median plane of the cylinder, 
the reticle will be seen to divide into two, and if the eye is inclined 
to the right or the left, the reticle will be bordered on one side by a 
black band of greater or less breadth, and on the other side by a 
bright band, this phenomenon being produced by the total reflexion 
which the luminous rays experience in traversing the cylinder obliquely 
and meeting the layer of balsam whose refractive index is less than that 
of the flint. 
