and Elementary Theorems of Geometry. 91 



Pa.QC and Pa.QC the required limits. And it is evident 

 these limiting values have like signs, and that according as 

 any value ofm.n (of the same sign) is comprehended between 

 them, equal to one of them, or not comprehended between, 

 so accordingly will the corresponding circle AIKB cut MM 

 in two imaginary, in two real and coincident, or in two 

 real and distinct points, E, &c. If B and K be not on the 

 same side of MM, the limits are imaginary, and therefore the 

 points E always real. 



Remarks on the consequences arising from supposing particular 

 positions and magnitudes for the involved data. 



First. — If the point B were coincident with C, the rest 

 being as general as may be desired. Here the circle COB is 

 infinitely small, and the points I and O are coincident with 

 C. It is also evident that the direction CI is, in this case, 

 on a tangent to the circle AHC at C, and that the direction 

 of BI is determinable, because angle IC right to B ■= angle 

 8 right. It is also evident the circle AIB, having the zero 

 chord IB, touches the direction BI in B, and is therefore 

 determinable, &c. 



Secondly. — If B coincides with C, and that angle right 

 = zero, the rest being as general as may be. In this case it 

 is evident that CB may take any direction through C, and 

 therefore that the point I is that in which any straight line 

 " through C cuts circle CHA. Hence circle IAB coin- 

 cides with AHC, and the points E are those in which the 

 circle AHC intersects MM. 



Thirdly.— If MM and CQ are parallels, then QC and HA 

 are parallels, and the point H is at infinity, and therefore 

 the straight line CA lies entirely in the circumference CAH, 

 and gives the point I by its intersection with circle CBO. 

 And it may be further remarked that should the angle 9 

 right = zero, then will the straight line CB lie wholly in 

 circumference CBO, and give the point I by its intersection 

 with CA. 



Fourthly. — If the points I and B coincide, the circles 

 AIB, CBO, touch in B. And if I and C coincide, the 

 circles AHC, CBO, touch in C. 



Fifthly. — If the points B and A coincide, then the straight 

 lines EBO EAD coincide, and the point O coincides with D, 

 and the circle CBO with CAD. In this case the points I 

 are evidently innumerable, and occupy all positions in the 

 circumference CHB, and the problem becomes ' porismatic,' 



