Chrystal's equation is a first-order ordinary differential equation quadratic in the derivative 
of 
,
 |
(1)
|
where 
, 
, and 
are constants. When 
and 
, it reduces to Clairaut's
differential equation 
. Substituting
gives a solution whenever
 |
(2)
|
For 
, setting the discriminant
of the quadratic in 
to zero gives the curve
 |
(3)
|
The repeated root is 
. The curve
solves the equation on a nontrivial interval only when
its derivative agrees with that root, which is equivalent
to 
. This illustrates the distinction between a
discriminant locus and an actual solution (Chrystal 1897).
-Discriminant of a Differential Equation of the First Order,
and on Certain Points in the General Theory of Envelopes Connected Therewith."
Trans. Roy. Soc. Edinburgh 38, 803-824, 1897.