73 The Three Sections, Tangencies, 



centre O coincides with the double point / and the locus is an 

 infinitely small circle. 



When the circle P'C'D' cuts AB in real distinct point / and 

 /; it is to be remarked that these points are both between A 

 and Bj or both on the same side of A and B^ according as 

 the distance between A and B is greater than the sum or less 

 than the difference of s and t ; moreover^ when the distance 

 between A and B is less than the difference of s and t, the 

 two points / and / will lie in the production of AB through 

 A when s is less than t, and in the production through B 

 when t is less than s. And it is obvious that the locus is real 

 for any value of ^^ to which the corresponding centre O does 

 not lie between/ and/. 



Hence, when AB is greater than s + t, the locus is real for 

 all positive values of "- ; and_, for negative values of ^^, the 

 locus is a real finite circle, a real infinitely small circle, or an 

 imaginary circle, according as ^ is not comprehended between 

 the limiting values ^ and ^ or equals one of them, or is 

 comprehended between them. 



Similarly, when AB is less than the difference of s and t, it 

 is evident the locus is real for all negative values of ^*; and 

 for positive values of ^ the locus is real and finite, real and 

 infinitely small, or imaginary, according as -^ is not compre- 

 hended between, or equal to one of, or is comprehended 

 between the limiting values ^ and ^• 



In the casein which c.d = zero, the point D' coincides with 

 B and the circle CT'D' cuts AB in B and in another point/ 

 wliich lies between A and B, or coincides with B, or lies in 

 the production of AB through B according as s is less than, 

 or equal to, or greater than AB, and therefore the limits are 

 a known negative value and infinity negative when s is less 

 than AB, and a known positive value and positive infinity 

 when s is greater than AB. .-. when s is less than AB the 

 locus is real for all positive values of ■^, and also for the nega- 

 tive values of — which are not comprehended between the 

 negative limit and negative infinity ; and when s is greater 

 than AB the locus is real for all negative values of ~, and for 

 the positive values of -^ not comprehended between the posi- 

 tive limit and positive infinity ; and when s = AB the locus 



