160] THE GEOMETRY OF THE CYLINDROID. 153
while, if 0 be the angle XOT, and OT be denoted by r, we have
x = r sin (0 — a),
or x = y' cos a — x sin a ................(ii) ;
but, obviously,
OR = r cos (0 — a) + MT;
whence h = x' cos a + y sin a + z' cot ß ...........(iii).
Solving the equations (i), (ii), (iii), we obtain
x' = — x sin a + y cos ß cos a,
y' = + x cos a + y cos ß sin a,
z' = h tan ß — y sin. ß.
It appears from these that
x2 + y'2 = x- + y2 cos2ß,
x'y = xy cos ß cos 2a + (;y- cos2 ß — x2) sin a cos a.
The equation of the cylindroid gives
y (a/2 + y'2) — 2mxy';
whence we deduce, as the required equation of the section,
(A tan ß — y sin ß) (x2 + y2 cos2 ß)
= 2mxy cos ß cos 2a + 2m sin a cos a (y2 cos2 ß — x2);
or, arranging the terms,
sin ß cos2 ßy3 + sin ßyx2 — (m sin 2a + h tan ß) x2 + 2mxy cos ß cos 2a
+ (rø sin 2a cos2 ß — h sin ß cos /3) y2 = 0.
It is often convenient to use the expressions
x = h tan (0 — a) — m sin 2# cot ß tan (0 — a),
y^hsecß — m sin 20 cosec ß,
from which, if 0 be eliminated, the same equation for the cubic is obtained;
or, still more concisely, we may write
x = y cos ß tan (0 — a),
y = h sec ß — m sin 20 cosec ß.
This cubic has one real asymptote, the equation of which is
y sin ß = m sin 2a + h tan ß,
and the asymptote cuts the curve in the finite point for which
x = — tan 2a (h + m sin 2a cot ß).
The value of 0 at this point is — a.