409 



rity of triangles is sufficient to express the rule of derivation 

 of the point Q from P. 



3. Hence, attending still to signs of angles, we may see 

 (even without referring to the figure) that 



CQA = PAD, CQB = PBD ; 

 and that therefore 



AQB = CQB - CQA = PBD - PAD 



= (ADP + DP A) - (BDP+DPB) 

 = ADB - APB = ACB - APB ; 

 or that 



APB + AQB = ACB. (2) 



The sum of the two angles subtended by the fixed chord AB, 

 at the assumed point P and at the derived point Q, is there- 

 fore constant, and equal to the angle which the same chord 

 subtends at the point C (or D) ; these angles being supposed 

 to change their signs when their vertices cross that fixed 

 chord. (This result was given in the Philosophical Magazine 

 for February, 1853, as one of several which had been obtained 

 by applying quaternions to the question.) 



4. In like manner if we continue to derive successively 

 other points, R, S, T, U, . . . we shall have 



AQB + ABB = ACB, &c, 

 and therefore 



APB = ABB = ATB = .. 

 AQB =ASB =AUB=.. 



the alternate points, P, R, T, ... are therefore situated on 

 one circular locus (-M), and the other alternate and derived 

 points Q, S, U, . . . are on another circular locus (N) ; or ra- 

 ther these two sets of alternate points are contained on two 

 circular segments, both resting on the fixed chord AB as their 

 common base (as stated in the Phil. Mag. just cited). 



5. It is evident also that if E, Fbe (as in fig. 1) the sum- 

 mits of the two semicircles on CD, of which the former con- 



} (3) 



