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.

208 DR TAYLOR, THE GEOMETRY OF KEPLER AND NEWTON.
it stands suppose DBo drawn to PQ. Then, the points A, B, C, P being fixed and
D variable,
{0} = {R} = T} = {{o}.
But P is the position of 0, and likewise of co, when RT vanishes. Therefore 0w
passes through some fixed point F. When D is at A the line Oo becomes QS, and when
RT passes through B it becomes CH. Thus F is the point (CH, QS), and as Ow turns
about F the point D is found by drawing CO, Bw.
Cor. 12. By the construction of Cor. 7, as is well known, we can describe the
conic through five given points. For example, in the limiting case in which three
points A, B, C and the tangents at B, C are given, we can take AB, A C for the
fixed lines, and for the fixed points B, C and the intersection Z of the two tangents.
LEMMA A. To find the centre of an involution of four points.
To find the centre of the involution in which P, K and S, T are conjugate points,
through P and S (or T) draw parallels, and through T (or S) and K draw parallels
meeting them in R and C respectively. Then RC passes through the centre of the
involution (AMGC, p. 258). The converse has in effect been used in Prop. A, where
the conic and AC, BD cut a parallel to AB in points of an involution having O for
centre.
The six joins of any four points cross any transversal in three pairs of points in
involution. In the above construction two of the four points are at infinity.
9.
Locus AD TRES ET QUATUOR LINEAS.
APOLLONIUS OF PERGA. We shall see that Newton mentions Apollonius of Perga
in connexion with the problem of the quadrilinear locus. What Apollonius says of the
707roÇ e7r TpEtS Kat reéo-apas 7pax/aaç is translated as follows by Dr T. L. Heath in
his edition of the Conics of Apollonius in modern notation (p. lxx. sq., 1896), "Now
of the eîght books the first four form an elementary introduction;...The third book
contains many remarkable theorems useful for the synthesis of solid loci and determinations
of limits; the most and prettiest of these theorems are new, and, when I had discovered them, I observed that Euclid had not worked out the synthesis of the locus
with respect to three and four lines, but only a chance portion of it and that not
successfully; for it was not possible that the synthesis could have been completed
without my additional discoveries." This prepares us to find in the third book of the
Conics of Apollonius, if not the synthesis of the locus, the elementary theorems on
which it depends.
Turning to lib. III. 54, 56 we see the property of the locus proved incidentally for
the case of three lines in the proposition thus enunciated by Dr Heath (Prop. 75, p. 120),