A sequel to the first six books of the Elements of Euclid, containing an easy introduction to modern geometry, with numerous examples. By John Casey.

GENERAL THEORY OF ASSOCIATED FIGURES. 223

nouvelles e'tudes de geometrie du triangle." The following
passage, in a note by Mr. M'Cay in Tucker's Paper, shows the
idea of extension had occurred. to that geometer:-" I believe
all these results would hold for a polygon in a circle, if the side
were so related that there existed a point whose distances from
the sides were proportional to the sides."-Mcarch, 1886.
Since the date of the foregoing note, which appeared in the 4th
Edition of the " Sequel," two Papers on the Harmonic Polygon
have been published; one by MM. NEIJBERG and TARRYCongress of Nancy, 1856, Association Pranfaise _pour l'avancemeat des Sciences; the other by the REv. T. C. SIMMoNs-Proceedings of the london Mat hematical Society, April, 1887. I am
indebted to the former of these for the demonstration of Prop. 2,
and to the latter for the enunciation of Prop. 7. With these
exceptions, and Prop. 13, all that is contained in this Section is
original.
SECTION YII.
GENERAL THEBoRY OF ASSOCIATED FIGURES.
PEF. i.-If any _point X in the circumference of a circle
Z be j~oined to n ftxed points I,, I2,... I,,, on the same
circumference, and portions IA,, IA,... IA,, be
taken on the joining lines in given ratios d,, d,2... d
and all measured in the same direction with res~pect to
a, system of figures directly similar, described on
I 1I 2A2. I, II,A,,, is called an associated system.
PEF. ii.-The _points I,, I, I,, are called the invariable points of the, system,