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. 227
is the circle of similitude. Join A,3. This will meet
Z in I3, which will be the invariable point corresponding to the next side, AA4, of the Tarry's line. Join
f'I3, and produce to meet the line A3A4, making with
QA3 an angle equal to o in A4. We construct in the

same manner A4A,, AA,, &c. Tarry's line, it is obvious, may be continued in the opposite sense, AAA,.
Cor.-If one side of a Tarry's line and its Brocard
points be given, it may be constructed.
Prop 8.-If the Brocard angles w, o' of a Tarry's line
be equal, the points Al, A2.. A, form the vertices of a
harmonic polygon.
Dem.-From the invariable points Ii, I2, &c., let
fall perpendiculars IC,, I2C2, &c., on the lines AA2,
A2,A, &c. Then, since these perpendiculars are homologous lines of F,, F., they meet on Z. Let 0 be their
point of intersection. Now, since co = o', the triangle