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 + DPA) - (BDP+ DPB) 
= ADB - APB = ACB - APB; 
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, 8, T, U, ... we shall have 
AQB + ARB = ACB, &c., 
or that 
and therefore 
APB= ARB=ATB=-..., (3) 
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 Z, F be (as in fig. 1) the sum- 
mits of the two semicircles on CD, of which the former con- 
