76 The Three Sections, Tangencies, 



AB. And it is obvious tliat for all negative values of -^ (as tlie 

 corresponding points O lie between A and B) the locus is ima- 

 ginary ; and it is further evident that for positive values of ^* 

 between zero and the lesser limits and between infinity and 

 the greater limits the locus is also imaginary ■ and that for all 

 other positive values of -^ the locus is real. And for a value 

 of — equal either limit, the locus is infinitely small. 



Having remarked the principal features of these three 

 divisions, the problem can now be considered under a 

 different form. For as the relations of the involved data 

 were expressed by 



AP2 + a.b : BP^ + c.d : :m : n 

 they are evidently expressed by 



n (AP2) - m (BP2) = m {c.d) - n {a.b) 

 and this can be orally expressed in the two following 

 manners; just according as we suppose -^ positive or negative, 

 viz : — 



First enunciation — (when -^ is positive) — " Given two points 

 A and B in position ; to find the locus of a point P, such that 

 n times the square of its distance from A, minus m times the 

 square of its distance from B shall be equal to a given magni- 

 tude g.h/^ regarding n and m as real numbers. 



Second enunciation — (when -^ is negative) — " Given two 

 points A and B ; to find the locus of a point P, such that n 

 times the square of its distance from A, plus m times the square 

 of its distance from B, shall be equal to a given magnitude g.h," 

 regarding n and m as real numbers. 



It is evident we can always determine a.b and c.d so that 

 m.{cd)—n {ab) = gh, or, better still, we can always find a.b 

 such that -n {a.b) - g.h, and then regarding cd = zero, we 

 still have m{c.d) -nfa.b) = g.h, and 



AP2 + a.b : BP^ : : m : n. 



In this form it is evident (from the preceding part of the 

 discussion) that one of the points / coincides with B, and 

 that the other point /satisfies the relation /A. AB = a.b. 



Now, when ^ is negative (and O between A and B) as we 

 can always assume m negative, and n positive, it is evident 

 that as w(AP2 — ) ^(BP^) consists of two necessarily positive 

 terms, the g.h in the second of the two preceding enuncia- 

 tions, can be regarded as always necessarily positive. This 



