and Loci of Apollonius, 8^c. 71 



AP.CP = QP.DP. But QP.DP : BP.DP : : QP : BP : : 

 AO : BO : : m : w ; hence AP.CP : BP.DP : : m : n, 

 and since AP.CP = AP^ + a.b, and BP.DP = BP^ + c.d, 

 therefore AP^ -t- a.b : BP2 + c.d :: m : n. 



DISCUSSION 



It is evident the locus is real only when the centre and 

 radius are real; and it is also evident the centre is real for 

 all real values of j^, and that the radius is real only when the 

 point O does not lie within the circumference P'C'D'. 



It is evident the circle P'C'D' will cut AB in real points 

 when a.b and c.d are both positive, or when one of them is 

 positive and the other negative ; but when a.b and c.d are 

 both negative, as the points P', C, D' are then on the 

 same side of AB, it is obvious the circle P'C'D' will or will 

 not cut AB in real points according to the conditions of the 

 data. 



And if / and / represent the points in wliich the circle 

 P'D'C cuts AB, it is evident that /A. A/ := ab, and that 

 /B.B/ = cd, the equalities being in signs as Avell as magni- 

 tudes. 



When a.b and c.d are both negative. 



Putting s and t to represent the magnitudes of mean pro- 

 portionals between a and b, and c and d, it is endent that 

 when the distance between the given points (disregarding 

 directions) is less than the sum of s and t, or greater than 

 their difference, the circle P'C'D' will not cut AB in real 

 points ; and when the distance between A and B is equal to 

 either the sum or difference of s and t, the circle P'C'D' Tvill 

 cut AB in real coincident points ; and when this distance is 

 greater than the sum of s and t or less than their difference, 

 the circle P'C'D' -will cut AB in real distinct points /, /. 



When the circle P'C'D' does not cut AB in real points, it 

 is evident the tangent OP' is real, and therefore that the 

 locus is a real circle for all real values of ^ positive or 

 negative. 



When the cii'cle P'C'D' touches AB, the tangent OP' is 

 ahvays real, and in this case too the locus is always real : 

 moreover, it is evident that when m : n : : Af : B/, the 



