Invariant Theory of Conies. 483

386. The invariants ®1? ®3 for the Brocard ellipse

+ \/ xzfaz = 0,

V

and the Jerabek hyperbola

sin 2Al sin (Az - A^x^ + sin 2A2 sin (A9 -

+ sin 2^3 sin (^ - ^fo)^^ = 0
are all to a factor

cos Al sin (A2 - Az} + cos A2 sin (-43 - ^x) + cos Az sin (^j - A2}
and its square, each of which is equal to zero. Hence the
Brocard ellipse, the Jerabek hyperbola, and their coyariant F
form a harmonic system.
The coyariant F is

.

_ _

(^22 - #32) sin 2-4! (^32 - #i2) sin 2-42 (^2 - <z22) sin 2^

(995)

EXERCISES.

1. Find the conic which forms a harmonic system with any two of Artzt's
parabolas, whose equations in harycentric co-ordinates are

x i2 =

and prove that it is a hyperbola.
2. The conic

Va?i sin (-4i - 0) + Va?2 sin (^2 - 0) + Va;3 sin (^t3 - 0) = 0,
Kiepert's hyperbola, and
a;i2 sin (A i — 6) sin (A% — A$ + #22 sin (^2 — 0) sin (As — ^i)
4- #s2 sin (-4s — 0) sin (A\ — -^2) = 0
form a harmonic system.
3. The incircle, the hyperbola, which is the isogonal transformation of
the right line passing through the incentre and circumcentre, and the
parabola
aixrl(a% - 03) + «2«22/(«3 - «
form a harmonic system.
O -r O