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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 FQ', CR' in another 

 point thereon, and so on for ever : and the same thing holds 

 for the lines DP', -FQ', 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 PVQ or QVR' 

 will be constant, and equal to VFW; so that if this latter angle 

 be commensurable with a right angle, the points P Q'22' . . ., 

 and therefore also the points P QR . . . 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. 



• Note, added during printing Since the foregoing communication was 



made, the author has seen how to ohtain such geometrical proofs, or confir- 

 mations, of all the foregoing results. 



