and Elementary Theorems of Geometry. 85 



Limits for the ration -• 



To find the limiting values for the ratio -, when the rest 

 of the data is unchangeable, we may proceed as follows : — 



The points C and H are fixed independent of the varia- 

 tions of -. And our object is to determine the limiting 

 positions for A on PA (which is fixed in position) for then 

 will the resulting values of ^ be the limiting values required. 

 Now the angle EA right to B being evidently constant, it 

 follows that the other point K, in which PA again cuts circle 

 AEB, is constant, no matter how A may vary ; and there- 

 fore putting a and a for the two points in which the two 

 circles through B and K to touch MM again cut PA, it is 

 evident that ~ and ~ are the required limits. It is clear 

 that these limiting ratios have the same sign, and that ac- 

 cording as any value of ™ (having like sign with these limits) 

 is of a magnitude comprehended between them — equal to one 

 of them — or not comprehended between them, so accordingly 

 will the corresponding circle AIKB cut MM in two ima- 

 ginary, in two real and coincident, or in two real and distinct 

 points, E. When ™ and the limiting ratios have different 

 signs, the points E are always real. If B and K be not on 

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

 points E real for all possible values of ~. 



Remarks on the consequences arising from supposing particular 

 positions and magnitudes for the involved data. 



If the point B coincides with C, then 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 a 

 tangent to the circle AHC at C, and that the direction BI 

 makes the angle IB right to C = angle ' right/ More- 

 over the circle AIB, having BI an infinitely small chord, is 

 touched by BI at B or C, &c. 



Secondly. — If MM and NN are parallels, then QC and 

 PA are parallels, and the point H is at infinity ; and there- 

 fore the straight line CA lies entirely in its circumference, 

 and gives the point I by its intersection with circle COB or 

 with straight line CB when angle ' 8 right ' is equal zero. 



Thirdly. — If B coincides with C, and that angle 6 equals 



