410 



tains the chord, and if M, N be the centres of the two loci, 

 then 



APB = AMI, AQB = AN1, ACB = ALI = 2 AFI; 

 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 (L), or its diameter EF, and we 

 may write 



LM.LN=LF>. (5) 



6. The similar triangles (1) give 



QA^QC QB = QC 

 PA JDA> PB DB' 

 and therefore 



QB.PA DA CB ... 



QArpB = m = m = C0 ^ (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 A 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 G and .ffbe summits of the loci (M) and (iV), so 

 chosen that the lines PG and QH, crossing the fixed chord 

 AB in the points F and Q, are both internal or both external 



