Dr. Brewster on the laws of polarisation, &c. a 6g 
refraction, varied in the same ratio as the polarising force, and 
that all the phenomena, whatever be the number of axes by 
which they are produced, may be calculated by the same 
general law which we have already established for the phe- 
nomena of polarisation. 
Let it be required, for example, to determine the velocity 
of the extraordinary ray in a crystal with any number of 
positive and negative axes. By the principles explained in 
Sect. IV. these axes may be reduced to two equivalent rect- 
angular axes, which may be either of the same, or of oppo- 
site names. Let us then take b = the axis of revolution of 
the two spheroids, a, a" the other axis, /3, /3 # the inclination of 
the incident ray to the axes of the crystal, ^ the angle of the 
forces as found in Sect. III. and f half the difference of the 
angles at the base of the parallelogram of forces. Then since 
the velocity of the light is inversely as the variable radius of 
the spheroid will be the square of the velocity of the ordi- 
nary ray, and i ~ the square of the minimum velocity of 
the extraordinary ray in virtue of the separate action of each 
axis. The difference between the squares of the velocities of 
the ordinary and extraordinary rays will be 
the sign being positive, when the axis is positive, and vice 
versa. But as these expressions represent the sides of the 
parallelogram of forces, we have 
Tang, f = 
a x b l 
Sin.*/3 — 
Sin.* /3'j Tang \ $ 
Sin. 1 /2 # 
a z ± b* c . 2 . 
Sm> 0 + 
