find Loci of Apollonius, ^c. 33 



spectively to tlic angles llA and S A right to N ; describe the 

 circle PGQ, and in it find O (on the same or opposite sides 

 of PQ with G according as |[: and H^; have like or un- 

 like signs) such that PO : Qo' : : /.RA : A'.SA ; draw OP 

 to cut AR in H ; describe the circle OAH : either point C 

 in which it cuts jNIN is an answerable point. 



Let E and F l)e the points in which GP and GQ cut AC. 



It is evident PE is parallel RA, and QF to SA, and 

 therefore that PE.S A : QF.RA : : PC.SC : QC.liC. Again, 

 the angle EP right to C being equal AR or AH right to C, 

 it is equal OH or OP right to C ; therefore a circle can pass 

 through OCPE, and the angle EC right to O = PC or PQ 

 right to O = GQ right to O ; hence, a circle can pass through 

 GFOE, and therefore (see Porism 4, Transactions for 1859), 

 PE : QF : : PO : QO : : /.RA : A'.SA, and therefore 

 PE.SA •• QF.RA : : I : k, and consequently PC.SC : 

 QC.RC :: / : k. 



DISCUSSIOX. 



It is evident the point of intersection I of OQ and SA is in 

 the circumference OKA (for the angle IQ or 10 right to A 

 = QO right to G = PO right to G, and .-. = IIO right to 

 A). ^ _ 



When f. is (as is supposed in the enunciation) confined to 

 a particular sign, there is but one point O, one circle AHO, 

 and therefore two (and but two) answerable points P — both 

 real or both imaginary. 



But if j: were unrestricted in sign, it is evident there would 

 l)e two points O, and therefore two circles OHA, and foiu* 

 points C. ]Moreovcr, since the points O must be on different 

 sides of j\IN, t\ro of these points C must be always real. 



Limiting Values for the Ratio •^• 



When the segments PS and QR lie partly on each other, 

 the points II and I lie on opposite sides of MN, and therefore 

 the corresponding point C must be ahvays real. 



Wheu one of the segments PS and QR lies entirely on the 

 other, it is evident the points A and G are on the same side 

 of ]MN, and therefore it is only when the ratio y. is positive, 

 that O, H and Acan be on the same side of MN; in otherwords, 

 the points C are real for all real negative values of 4 ; but 



for positive values of | the points C arc real only wheu the 



