38 The Three Sections, Tangencies, 



there are two answerable points O, and therefore two circles 

 OHAj and four points C. Moreover, the points O being 

 necessarily on opposite sides of PQ,, it folloAvs that two of these 

 four points C must be always real. 



Limiting Values for Right. * 



There are evidently limits to right only when A and O 

 are on the same side of MM. 



Now the point O is found independent of the magnitude of 

 6; and it is evident that by describing the two circles through 

 A and O which touch MM, and putting H' and H' for the 

 points in which OP cuts them again^ and T and I' for the 

 points in which AH' and AH' cut RB, then will the angles 

 I'A right to B and I'A right to B be the limiting values 

 for the angles right. And if X be the point in which AO 

 cuts I'I'j it is evident that according as ^y^-, and ~ have like 

 or unlike signs, so will any straight line through A cutting 

 I'l' and H'H' in L and H give ^' and ^^' of like or unlike 

 sign. Hence it follows that when ^ and ^, have like signs, 

 the limiting values include between them all values (and no 

 others) of the angle right^ for which AI and BI are imaginary ; 

 and when ~, and ^ have unlike signs, the limiting values 

 have outside them all values (and no others) of angle right, 

 for which AI and BI are imaginary. And when 6 right is 

 equal either of the Kmiting values, the lines AI are coincident 

 and real. 



Limiting Values for the Ratio j,^ 



When the points C may be imaginary, the point O must 

 evidently fall on the same side of PQ with the points A and 



H. And it is evident we arrive at the limiting values of ^^ 

 or QoT^, or (which is equivalent) of |^^ by finding the points 

 o' and o' in the circle PGQ, so that ?t' and h' being the points 

 in which Po' and Po' cut AL, the circles Ah'o' and Ah'o' 

 touch MM. 



Moreover, it is evident that the two ratios gg', and ^JJ 

 are the required limits, and that they have the same sign, and 

 that according as any value of | (having like sign) is of a 



.t,^*^"™.^*^^'^ ^^'^*' *^" ^^°* adopt my improved methods of indicating angles, 

 will find it impossible to define the limits of the angular magnitude 0. 



