and Loci of Apollonius, S^c. 75 



production of AB through A. Hence, we see that the limits 

 ^ and ^ are the one positive and the other negative. 



And it is e^•ident tliat for all positive \'alues of -^ the locus 

 is real, or real and infinitely small, or imagiuarj'^, according 

 as -^ is greater than the positive limit g^, or equal to it, or 

 less than it. And for negative values of ^ it is evident the 

 locus is real, real and infinitely small, or imaginary, according 

 as -^ is numerically greater than the negative limit ^^ or 

 equal to it, or less than it. 



If c.d = zero ; one of the points/ coincides with B, and one 

 of the limits = infinity ; and the other point / is in the pro- 

 duction of BA through A. In this case the locus is imaginary 

 for all finite negative values of '^, and real only for positive 

 values of — which are not less than the other limit g^; and 

 when -^ is equal either limit, the locus is real and infinitely 

 small. 



If a. 6 and c.d be each equal zero ; then one point /coin- 

 cides with A, and the other with B, and the limits are + zero 

 and - infinity, and the locus is real only for finite values of 

 — which are positive. 



And, by similar considerations as were made use of when 

 a.b and c.d were regarded as negative, we are led to know 

 that a particular case of this problem may be enunciated as 

 follows : — " Given a point A and a straight line IK in position; 

 to find the locus of a point P, suc?i that AP^ + a given positive 

 magnitude a.b shall be equal to the rectangle under a given 

 length 2.m, and the perpendicular distance PK of the point P 

 to the given line IK. In which the sign of 2.m in respect to 

 the direction of PK is given." 



The solution may be worded exactly as in the similar case 

 when a.b and c.d were regarded as negative. But it may be 

 remarked that here the points /, /, are always real when 

 the data is real, and that they are equidistant from I, and on 

 opposite sides of A, &c., &c., 



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



It is evident the points C, D', and the point P' are on 

 opposite sides of AB, and that the circle PCD' always cuts 

 AB in real points / and /, one of which is in eacli production of 



