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. 207
draw the diameter of PK. Then draw the diameter through I, and find its vertices,
and those of the conjugate diameter.
Cor. 3. Hexagrammum Mysticum. The construction in Cor. 1 for two pairs of
parallel chords gives three pairs, AB and KP, AC and PH, BH and KC. Hence
Pascal's theorem for the case of parallels.
Cor. 4. Given parallel chords AB, KP and a fifth point G of a conic, a sixth
point D on the curve can be found as follows. Draw any parallel to CK meeting
PK in T and meeting the parallel through P to A G in R. Then BT, CR meet at
D on the conic.
Cor. 5. In this construction we may say that PR, PT are to be taken in a given
ratio equal to SC/SK. See below on Newton's Lemma xx.
Cor. 6. The locus of the point (BT, CR) in Cor. 4 is a conic through A, K, C, P, B.
Hence the following construction. Take fixed lines PR, PT; fixed points B, C; and
a fixed point Z at infinity. Then as the line ZRT turns about Z the point (BT, CR)
traces a conic through B and C. Obviously it will likewise trace a conic in the general
case when Z is not at infinity.
Cor. 7. In other words, the locus of the vertex D of a varying triangle RDT whose
base slides between fixed lines PR, PT, while its three sides pass through fixed points
B, C, Z respectively, is a conic. This may be shewn independently as follows. Draw CD
in any assumed direction, and find R, and then T, and then D. Thus one point D is
found on the line through C, and it is a single point of the locus. By drawing the
line BC we find that each of the points B, C is a single point of the locus. Thus CD
cuts it in two such points, and the locus is therefore of the second degree.
Cor. 8. The anharrmonic point-property of conics. In Cor. 4, as D varies, the parallel
RT to CK divides PR, PT proportionally, so that the cross ratios of R and T in any
four positions are equal to one another. Hence
B {D} = {T} = R} =C{D},
or any four points 1D of the conic are equi-cross with respect to B and C, which may be
any assumed fifth and sixth.
Cor. 9. Hence we can deduce the general case of Cor. 6.
Cor. 10. Locus ad quatuor lineas. By similar triangles, PR/PT and SC/SK are
equal ratios. Compounding with them other equal ratios we get
PR. PQ/PS. PT = SO. SA/SK. SP =f/g,
if f, g be the focal chords parallel to A, AB. See also below on Newton's Lemma xvII.
Cor. 11. The extension at the end of Cor. 6 follows frorn a simple transformation
of the figure by which the parallels RT are turned into convergents. In the figure as