25° Dr. Brewster on the laws of polarisation , &c. 
is equal to any number of equal or inequal axes of the same 
name placed at A, the sum of whose intensities is equal to the 
intensity of the axis at A, it will follow — 
3 d. That the effect of a single axis may be represented by 
any even number of equal axes of an opposite name, pro- 
vided in every pair of axes one of them is at right angles to 
the other, and all of them lie in the same plane perpendicular 
to the original axis. The intensity of each of the new axes 
must be to that of the single axis as - to 1 , n being the num- 
ber of axes. 
4 th. That the effect of a single axis may be represented 
by any number of equal pairs of axes of an opposite nature, 
provided the axes which compose each pair are equal and 
rectangular, and all of them lie in the same plane perpen- 
dicular to the original axis. The intensity of the axes which 
compose each pair may differ in any way, but ~ or half the 
sum 5 of the intensities of all the axes must be equal to the 
intensity of the single axis. 
In the preceding observations we have supposed the axes 
of the crystals to be rectangular, a supposition which is by no 
means rendered necessary by the phenomena. It is obvious 
indeed that the effect of a single axis cannot be represented by 
two axes that have any other inclination but that of o°orqo°; 
but we shall presently see that the effect of two rectangular 
axes may be represented by two equal and inclined axes. 
Let ABC, PI. xvi. fig. 10 , be a spherical triangle representing 
the eighth part of the sphere, and having all its angles right 
angles. Then if G is the pole of one of the resultant axes or 
diameters of no polarisation, the angular distance « (3 of the 
two inclined axes a, /3 must be such, that the angle a, G/3 is a 
