Memoirs presented to the Cambridge philosophical society on the occasion of the jubilee of Sir George Gabriel Stokes, bart., Hon. LL. D., Hon. SC. D., Lucasian professor.

DR TAYLOR, THE GEOMETRY OF KEPLER AND NEWTON. 209
TQ, TQ' being two tangents to a conic, and R any other point on it, if Qr, Q'r'
be drawn parallel respectively to TQ', TQ, and if Qr, Q'R meet in r and Q'r', QR
in r', then
Qr. Q'r': QQ'2 = (P V2: PT2) x (TQ. TQ': QV2),
where P is the point of contact of a tangent parallel to QQ'.
Dr Heath shews (p. 122 sq.) that this proposition and his next (lib. III. 55), for
tangents to one branch and two branches of a conic respectively, "give the property of
the three-line locus." The constan.cy of Qr. Q'r' being a corollary from the property of
the trilinear locus, we can of course work back from the latter to the former.
But more briefly, leaving out r, r', draw the tangents TQ, TQ' crossing any chord
RR' parallel to QQ' in K, K'.
Then, because the diameter through T bisects both KK' and RR', the intercepts
KR, K'R are equal, and likewise KR', K'R.
LI
D, /
T L Q K
Therefore RK.RK' (or KR.KR') varies as KQ2.
This is the trilinear theorem as proved by Apollonius.
Inflect RD to QQ' parallel to QT.
Then RK.RK' varies as RD2, and the theorem may be stated thus,
The distance of any point on a conic from a given chord varies as a mean proportional
to its distances from the tangents at the ends of the chord, each distance being parallel to
any given Une.
Apollonius does not enunciate the theorem, but he proves and uses it in the course of
his propositions mentioned above.
VOL. XVIII. 27