and Loci of Apollonins, h^c. 29 



be on opposite sides of MM, it is evident two of these points 

 C must be always real. 



Limiiwg Values for the Ratio ^' . 



When A and II arc on opposite sides of M]M, the corre- 

 sponding points C are always real ; but when A and II are on 

 the same side of ^l^l, the reality of the points C is dependent 

 on the position of O, or, which amounts to the same thing;, 

 on the value of '," . xVgain, since PO : RA : : PQ : RS, it 

 is evident that if O' and O' be the points in which the two 

 circles through, A and H touching MM again cut PH, then 

 will jj V and J-^ be the limiting values of j^ . INIoreover, it is 

 evident that, according as any value of ;|^ is comprehended 

 between these limits, or equal to one of them, or not compre- 

 hended between them, so will the corresponding points C be 

 imaginary, or real and coincident, or real and distinct. 



Forismalic Relations of the Data. 



It is evident the problem is indeterminate only when the 

 circle AOH is indeterminate. AYhen O coincides with A, 

 and that PO cuts AH, then II also coincides with A, and 

 the circle AOH is infinitely small ; but when O coincides with 

 A, and that PO and RA form one straight line, then it is 

 obvious that any circle touching this straight line in A is an 

 answerable circle AHO : therefore in this case there are in- 

 numerable answerable points C and lines CAD. The problem 

 under these last conditions (viz., when we have M]M parallel 

 to NN, and PRA a straight line, and the ratio - equal to 



1^) is said to be "porismatic^^ — any straight line CAD 

 through A being an answerable line. 



Remarks. 



1. When INOI and NN arc parallels, it is evident PO 

 and RA are parallels and that H is at infinity (when O 

 and A are not coincident), and the circle AOH infinitely 

 great. In this case the straight line AO, lying in the infinite 

 circumference, will give one point C in its intersection with 

 MM : the other point C is evidently at infinity on MM. 



3. In all cases QO and SA intersect in the circumference 

 of the circle AOH. 



