410 
tains the chord, and if M, N be the centres of the two loci, 
then 
APB =AMI,AQB = ANI, ACB=ALi-2Ar; 
so that, by (2), 
AFI- ANI= AMI- AFI, or FAN= MAF: (4) 
wherefore the centres M, N are harmonic conjugates with 
respect to the given circle (Z), or its diameter EF, and we 
may write 
IM.LN=LF. (5) 
6. The similar triangles (1) give 
Ai QC OP. OC 
mt Ds. PB DE 
and therefore 
eet ae = 22 = a = const., (6) 
(as stated in the same number of the Magazine). Hence (as 
there stated) the successively and directly derived points 
Q, R, S,... must tend indefinitely to coincide with the fixed 
point B, and in like manner the inversely derived points 
R, Q, P,... must tend indefinitely to coincidence with the 
other fixed point A, as the limits of their positions, on account 
of the geometrical progressions of the quotients of distances 
from those two fixed points, wherever the first point P or S of 
the direct or inverse derivation may be: unless it happen to 
be exactly at either of those two fixed points A and B, in 
which case the derivation will produce no change of place. 
(It might therefore be not too fanciful to say that 4 and B 
are respectively positions of unstable and stable equilibrium 
for the direct mode of derivation, but of stable and unstable 
for the inverse mode.) 
7. Let Gand H be summits of the loci (J/) and (N), so 
chosen that the lines PG and QH, crossing the fixed chord 
AB in the points P’ and Q’, are both internal or both external 
