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. 225

follows that the lines joining the n(n + 1)/2 intersections of Li, L,,... L, to K pass respectively through
the n (n + 1)/2 centres of similitude.
DEF. IV.-I shall call K the perspective centre of the
polygon formed by the lines LI, L,,... L,,.
Prop 4.-In an associated system of n figures, the
centre of similitude of any two polygons, G, G', each
formed by a system of n homologous lines, is a point on the
circle of similitude.
Dem.-Let K, K' be the perspective centres of G,
G': thus K, K' are corresponding points of G, G'. Let
I be any of the invariable points. Join IK, IK', and
let the joining lines meet any two corresponding lines
of G, G', in N, N'; then KN, K'N' are corresponding
lines of G, G'. Hence the centre of similitude is the
second intersection of the circumcircle of the triangle
INN' with the circle of similitude. Hence the proposition is proved.
Prop. 5.-Thesix centres of similitude of an associated
system offour figures taken in pairs are in involution.
Dem.-Let L, 1, L3, L4 be four homologous lines
of the figures, and K the perspective centre of the
figure formed by these lines. Then the pencil from K
to the six centres of perspective passes [Prop. 3]
through the three pairs of opposite intersections of the
sides of the quadrilateral LI, L,, L3, L4, and therefore
forms a pencil in involution.
DEF. v.-If in an associated system of n figures F1,
F... F, there exist n + 1 points A1, A2... A,, A,+1,
such that AA,, A2A,... A,,A,,+ are homologous lines
of the figures; then the broken line A,1A... AA,+1 is
called a TARRY'S LINE; and, if A,,,, coincides with Al, a
TARRY'S POLYGON.
I have named the line of this definition after M. Gaston Tarry,
"Receveur des contributions, a Alger," to whom the theory of