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radical axis, of themselves and of the given circle described on 
CD as diameter. III. The centres of the two alternate loci 
are harmonic conjugates with respect to the given circle. 
IV. If from two fixed summits of the two loci chords be drawn 
to the successive points, and prolonged (if necessary) till they 
meet the radical axis in other points P’, Q’, &c.; if also a sum- 
mit F of the given circle be suitably chosen (on the line of the 
three centres), then the two lines FP’, CQ’ will cross in one 
point on the given circle, the two lines /'Q’, CR’ in another 
point thereon, and so on for ever: and the same thing holds 
for the lines DP’, F'Q’, or D Q’, FR’, &c. Particular forms of 
these theorems have been published in the Phil. Mag. for this 
month (February, 1853), but only for the case when the top 
of the rectangle, or the radical axis, meets the given circle in 
two real points, A, B, in which case the derived points Q, R,.. 
converge towards the point B nearer to C. In the contrary 
case there can be no convergence, but there may be circulation 
in a period. For if we then denote by V one of the two com- 
mon points of the system of common orthogonals, and by W 
the point of contact of the given circle with a tangent drawn 
from the middle point between them, the angle PV Q’ or QV FR’ 
will be constant, and equal to VF W;; so that if this latter angle 
be commensurable with a right angle, the points P Q’R’..., 
and therefore also the points PQR ... will recur in a certain 
periodical order. These conclusions have been by Sir W. R. 
Hamilton obtained as results of his quaternion analysis ; but 
he believes that it will not be found difficult to confirm them 
by a purely geometrical process, founded on the known theory 
of homographic divisions. 
* Nore, added during printing.—Since the foregoing communication was 
made, the author has seen how to obtain such geometrical proofs, or confir- 
mations, of all the foregoing results. 
