crystallized bodies on homogeneous light. 79 
which determines the equation of any one of these curves, we 
must select a crystal, where the proximity of the axes and 
intensity of the polarising forces are such, as to bring the 
whole system of rings within a very small angular compass ; 
as by this means we avoid almost entirely the disturbing 
effect of the variation in thickness, arising from obliquity 
of incidence. Dr. Brewster, in his Paper of 1 818, has chosen 
nitre, as affording the best general view of the phenomena, 
and it is admirably adapted for this purpose ; the whole sys- 
tem of rings being comprised at a very moderate thickness 
within a space of io°, allowing us to regard their projection 
on a plane perpendicular to the optic axis as a true represen- 
tation of their figure, undistorted by refraction at the sur- 
face, &c. If we examine the rings in this crystal (illuminated 
with homogeneous light, or by the intervention of a red glass 
in common day-light) it will be evident that the general 
form of any one of them is a re-entering symmetrical oval, 
which no straight line can cut in more than four points, and 
which, by a variation of some constant parameter, is in one 
state wholly concave, as 1 (Fig. 4. Plate V.) then becomes 
flattened, as 2 ; then acquires a minimum ordinate and points 
of contrary flexure, as 3 ; then a node, as 4 ; after which it 
separates into two conjugate ovals, as 5 ; which ultimately 
contract themselves into the poles P, P' as conjugate points. 
The general idea bears a striking resemblance to the vari- 
ation in form of the curve of the fourth order, so well known 
to geometers under the name of the lemniscate, whose equa- 
tion is 
(x 2 -f /-[- a 2 ) 2 = a 1 (6 2 + 4 x 2 ) 
when the parameter b gradually diminishes from infinity to 
