The Imaginary in Geometry 53 



If we fix 0', while 6" is allowed to vary, the black point (x', y') 

 will move on the hyperbola 



( b"x' cos 6' + a"y' sin 0') 2 — ( &V sin ^ — a'/ cos 0') 2 



=> (a'6" cos 2 V + a"&' sin 2 (9) 2 , 



while the blue point (£,17) is at the same time moving on 



[ (6" — &')£ cos + (a" — a')*? sin 0'] 2 — [ (6' + &")£ sin 0' 



— (a' + a") v cos 6') ] 2 

 = [(a' + a")(b" — &0cos 2 r+ (a" — a') (&' + &") sin 2 0'] 2 . 



Rearranging, these become, respectively, 



(&" 2 cos 2 ^ — &' 2 sin 2 0')*'* + 2(0'^ + a"b") sin 0' cos B'x'y' 



+ (a" 2 sin 2 ^ — a' 2 cos 2 ^)y' 2 



= a'*b"* cos 2 0' + 2a'a"b'b" cos 2 sin 2 + a" 2 b' 2 sin 2 0', 

 and 



[ (b' 2 + &" 2 ) (cos 2 ^ — sin 2 0') — 2b'b"]e 



+ 4(a'b' + a"b") sin & cos 0'^ 



— [ (a' 2 + a" 2 ) (cos 2 0' — sin 2 0') + 2a'a"] v 2 



= [(a'b" — a" b') (cos 2 6' — sin 2 6') -\- a"b" —a'b'\ 2 . 



Considerable simplification results, of course, for 0' equal to an 

 integral multiple of 71-/2. 



If, on the other hand, we fix 6" while 0' is allowed to vary, the 

 black point will move on the ellipse 



{b"x' sh 0" -j- a'y' ch 0") 2 + {b'x' ch 6" — a"y' sh 0") 



= (a"b" sh 2 0" + »'&' ch- 0" ) 2 , 

 while the blue point is moving on 



[(&" — V)t sh 0" + (a' + a")i?ch 0"] 2 



+ [(b'-\-b")ich0"—(a" — a') v sh6"] 2 

 = [(a" — a')(b" — b') sh 2 0" + (a' + a") (b' + b") ch 2 0"] 2 . 



53 



