and Loci of Apollonius, S^-c. 73 



is real for all real values of ^. And in any of these three 

 cases it is evident the locus is an infinitely small circle when 

 -^ is equal either of the limiting ratios of the case. 



In case a.b = zero, and c.d = zero ; then the points C and 

 D' coincide with A and B, and the points/ and/ also coincide 

 with A and B. And it is evident the limits are equal zero 

 and infinity, and that according as any value ^ is positive 

 or negative, so ^^ ill tlie locus be real or imaginary ; when -^ 

 is equal either of the limits, the locus is an infinitely small 

 circle coincident with one of the given points. 



If the points A and B coincide, and that a.b = c.d, and that 

 ^ = + 1 ; then since the point O may have any position 

 whatever in the line through A and B, and that the circle 

 P'C'D' is real, it follows that there are innumerable circles 

 such that any point P in the circumference of any of them 

 fulfils the conditions of the locus. The problem in this state 

 of the data is said to be " porismatic.^^ 



When we suppose ^l = -\- I, the centre O of the locus is 

 infinitely distant (when the points A, B are distinct) and the 

 tangent to the circle P'C'D' is parallel to AB ; and it is evident 

 the perpendicular from the point of contact on AB is entirely 

 in the infinitely great circle constituting the locus, and 

 passes through a' the center of the circle P'C'D'. In this 

 case we have AP^ — BV^ = c.d — a.b, and the problem can be 

 enunciated as folloAvs : — Given two points A and B to find the 

 locus of a point P, such that the square of its distance from A, 

 one of the given points , minus the square of its distance from the 

 other, shall be of a given magnitude c.d - a.b of knovm sign. 



And if we suppose I a point in the line AB such that 

 BI.IB = c.d, and that I remains fixed when B is infinitely 

 distant ; then it is obvious that P'D' and PD are parallel to 

 AI and bisected by the pei'pendicular to AB through I ; and 

 it is evident that when -^ is of a finite magnitude, this perpen- 

 dicular through I is entirely in the infinite locus. 



Now it is further evident that if Ave suppose n always 

 := BI, and that B is at infinity in the direction AI, 

 then since AP.CP : BP.DP : : m : BI, and that for points 

 P at finite distances, BP = BI, therefore AP.CP = J)V.m. 

 And it is obvious that as DP is double the perpendicular from 

 P on the perpendicular through I to AB, the problem 

 becomes tantamount to what is given in the enunciation : — 



