und Loci of ApoUonius, ^c. 31 



O, and the iutL'i'scctiou II of RA and PO, describe a circle ; 

 through either point C in which this circle cuts jNOI draAv 

 CA to cut XX in D : then will CAD he an answerable line. 



For draw CO. 



The angle Oil or OP riiilit to C is = AH or AP right to 

 C or D ; and the angle PC) right to Q = All right to U : 

 therefore the angle CO right to Q is equal AD right to U ; 

 hence the triangles ADU, COQ arc similar, and QC.T'D = 

 QO.UA. But the similar triangles PQO, AUR give QO.UA 

 == QP.rP ; therehne OC.UD = QP.UR = m.n. 



WISCUSSIOX. 



AYhenwi.M is restricted in sign (as in the enunciation), there 

 is evidently but one ansAvevable point O, and therefore but 

 one circle AOH, and two points C, real or unreal, according 

 as the circle AOH cuts ^sVSi in real or imaginary points, 



If m.n be unrestricted as to sign, then there are evidently 

 two answerable points O, and therefore tAvo circles AOH, and 

 four points C, and lines CAD : moreover, it is evident that 

 the points O are on opposite sides of M]\I, and therefore that 

 two of the points C must be always real. 



Limiting Values for m.n. 



It is evident the points C can be imaginary only v.hen 

 A, O, and H are on the same side of jMM. We know one 

 point A in the circle AOIIC, but, in order to arrive in a sim- 

 ple manner at the limiting positions for the circle AOH, it 

 M'oidd be well if Ave could find another point in the circum- 

 ference. "We can find such a point. For, if T be the point in 

 Avhich QO again cuts the circle, the angle AH right to T = 

 OH right to T, and is therefore = RA or RH right to X ; 

 and hence AT is parallel to XX, and the point T in which it 

 cuts QO is known. 



X'oAv it is evident that by putting O' and O' for the points 

 in which the circles through A and T, touching jIM, cut 

 OO, then will X'A.QO' a.ud UA.QO' be the required limits. 

 Moreover, it is evident that according as any A-alue oi m.n is 

 comprehended between these limits, or equal to one of them, 

 or not comprehended between them, so will the correspond- 

 ing points C be imaginary, or real and coincident, or real and 

 distinct. 



Porismatic Relations of the Data. 



It is evident the problem becomes indeterminate only when 

 the circle AOH becomes indeterminate. X'ow the circle 



