270 Dr. Brewster on the laws of polarisation, &c. 
Consequently the difference between the squares of the 
velocities of the ordinar)^ and extraordinary ray produced by 
the combined action of the two axes will be 
(fl±L‘ S in.*/3) (Sin. 4 ) 
Sin. (C+ 2 
Hence, calling V the velocity required, we have 
v*=F±- 
a* b z 
Sin. 2 /3) (Sin. %{/) 
Sin. (£+±*) 
± b z 
and 
V = i T,± 
Sin. 2 /3) (Sin. 
> 
Sin.(CH-i'l') 
The form of the compound, or irregular spheroid, may 
therefore be computed for all doubly refracting crystals. 
The general law of double refraction which has now been 
explained, may be thus expressed. 
The increment of the square of the velocity of the extraordinary 
ray produced by the action of two axes of double refraction , is equal 
to the diagonal of a parallelogram whose sides are the increments 
of the square of the velocity produced by each axis separately , 
and calculated by the law of Huygens, and whose angle is 
double of the angle formed by the two planes passing through the 
ray and the respective axes. 
When the two axes are of equal intensity and of the same 
character, the preceding law gives the very same results as 
the law of Huygens does for one axis placed at right angles 
to the other two. 
It is scarcely necessary to observe, to those who have 
studied the preceding sections, that the phenomena of double 
refraction cannot be referred to the ordinary action of at- 
