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.

BOOK VI.

125

the chords AB, A'B' to be indefinitely near, we can infer from it a remarkable property of the motion of a
particle in a vertical circle, and also a method of representing the amplitude of Elliptic Integrals of the First
kind by coaxal circles.'
Prop. 12.-PONCLETr'S THEOREM.-If a variable poTygon of any number of sides be inscribed in a circle of a
coaxal system, and if all the sides but one in every position
touch fixed circles of the system, that one also in every
position touches another fixed circle of the system.
It will be sufficient to prove this Theorem for the
case of a triangle, because from this simple case it is
easy to see that the Theorem for a polygon of any number of sides is an immediate consequence.
Let ABC be a A inscribed in a 0 of the system,
A'B'C' another position of the A, and let the sides AB,
A'B' be tangents to one ( of the system, BC, B'C'tangents to another 0; then it is required to prove that
CA, C'A' will be tangents to a third 0 of the system.
Dem.-Let the perpendiculars from A, B, C on the
radical axis be denoted by p, p', p", and the perpendiculars from A', B', C' by mr, r-', ir"; then, by Prop. 11,
we have
AA': BB':: /p -- \/- /-p + V/~ '
and BB': CC':: P' + -v ': VP" + V/";.. AA': CC':: Vr/ + r: l/y' + v/7.
Hence AC, A'C are tangents to another circle of the
system.
The foregoing proof of this celebrated theorem was
given by me in 1858 in a letter to the Rev. R. Townsend, F.T.C.D. It is virtually the same as Dr. Hart's
proof, published in 1857 in the Quarterly Journal of
Mfathematics, of which I was not aware at the time.
* The method of representing the amplitude of Elliptic Integrals by coaxal circles was first given by Jacobi, Crelle's Journal,
Band. III. Theorem 11 affords a very simple proof of this application. See Educa;tional Times, Vol. II., Reprint, page 42.