The Imaginary in Geometry 57 



-f [ (a' 2 + a" 2 ) (ch 2 9" + sh 2 0" ) — 2aV] sh 2 <£} 

 + 2$ v {2(a'b" — a"b') ch (9" sh 5" 

 _j_ ( a " a _<*'*_&'« + b'*) ch </> sh </>} 



+ >/ 2 { [ («" + a" 2 ) (ch 2 0" + sh 2 0") + 2oV] ch 2 </> 

 + 4(a'b' + a"&") ch 0" sh 0" ch <£ sh <£ 



+ [(&'■ + &"■) (ch 2 9" + sh 2 0") — 2W] sh 2 0] }. 



In these more general equations the coefficient of fy is no longer 

 a mere multiple of that of xy, and the condition that the square 

 of the discriminant of the left member should equal the opposite 

 of the constant term can be satisfied independently for the two 

 ellipses. The six unknowns a' , b' , a", b" , 9" , <f> are then connected 

 by six independent 1 equations. Thus any two ellipses with center 

 at origin and which will give real values for these unknowns can 

 belong to the system. 



In this most general case, as in the simpler ones formerly con- 

 sidered, we can have the black ellipse and the blue ellipse simul- 

 taneously described by motions which are composed of two simple 

 harmonic motions; the motions, in fact for 9' varying uniformly 

 with the time, are given by 



x> = cos 9' (a' ch 9" ch 4> + b' sh 9" sh <£) 



+ sin V (a" sh 9" ch <j> + b" ch 9 " sh <£•) , 



y' = cos 9' (— b" sh 9" ch <f> — a" ch 9" sh <£) 



+ sin 6' (&' ch 6" ch <£ -f a' sh 6" sh <f>) , 

 and 



£ = cos 9' (a' + a") ch 0" ch </> + (&' -f b") sh 6" sh <f>) 



+ sin & [ (a"— a') sh 9" ch 4> + (6" — 6') ch 9" sh c/>), 



r^costf' [(&' — b") sh#"ch</> + (a' — a")ch0"sh<£) 



-fsinfl' [(fc'-ffc") ch0"ch0 + (a' -fa") sh0"sh£). 



The elimination of 0' from these equations gives the foregoing 

 equations of the black ellipse and the corresponding blue ellipse. 

 If from the same equations we eliminate 9" we. get a pair of 

 hyperbolas. 



1 The Jacobian does not vanish, as can be seen by substituting therein 

 special values for the unknown. 



57 



