The Imaginary in Geometry 55 



in fig. 48, A, B, Q, R, T, U, are seen each to belong to two curve 

 pairs. 



The intersection with the line infinity is composed of the inter- 

 sections of the asymptote pairs which again are the two involu- 

 tions cut out by 



"■"2 ~2 



x y 



~2 + f§ = O. 



a b 



Notice that for the asymptotes of a black hyperbola to coin- 

 cide with the asymptotes of a corresponding blue hyperbola would 

 require that the coefficients of the left members were proportional 

 and so 



a'"- — a" 2 + 2a' a" = b' 2 — b" 2 + 2b' b" =0, 



which is independent of 0. Thus, if there is a single case of 

 coincidence, for a given locus, there is always coincidence. 



Not every pair of concentric ellipses can belong to such a sys- 

 tem as that determined by 



X" V 



a b 

 For in order that the pair 



A x x' 2 + 2Hx'y + B iy ' 2 = C*, A£ + ifl^ + Brf = C 2 2 , 



should belong to the system, it is necessary that simultaneously 

 we can have 



A 1 B 1 — H- = C 1 - and A,B. 2 — 4H- = C 2 2 . 



This imposes one condition upon the ellipses. That condition 

 satisfied, we have five equations in the five unknowns a', a", b' , 

 b" , 8" , the equations, namely, 



b' t ch 2 6" + b"*sh. 2 6" = A 1} 

 a"ch 2 0"-f a"*sh 2 6"=B v 



(a'b" — a"b') ch 0" sh 6" = H, 

 (b' 2 + b" 2 ) (ch 2 6" -f sir 0" ) -f 2b'b" = A 2 , 

 (a' 2 + a" 2 ) ( ch- 6" + sh 2 6") + 2a'a" = B 2 . 



55 



