56 Ellery W. Davis 



There is somewhat greater complication introduced if we sup- 

 pose the conic 



x y 



=2 + T2 = I 



a b 



to be turned through the imaginary angle i<f>. 

 We then write 



x cos icf> -\- y sin /</> = x ch 9 -f- iy sh 9 = a cos 

 and 



— .r sin i(f> -f- ^ cos 19 = — ijt sh <f> -\- y ch<j> = b sin 0, 



the equation of the conic becoming 



~ 2 / ch" <j> sh" p \ / I i \ 



re" I —^2 r>" I + 2 2xy sh 9 ch 9 I ~, — =r, I 



\ a b" ) \a b" / 



, ,.,/ -sh 2 9 , ch 2 9\ 



When now 0" is fixed we find, on elimination of 0', that (.v',3/') 

 moves on 



x' 2 { (&'*ch*0"H- &" 8 sh 2 0")ch 2 9 +2(a'&' + a"5")ch0"sh0"ch9sh9 

 + (a' 2 sh 2 6" + a" 2 ch 2 6") sh 2 0} 



-f 2.ry{ ( a 'b" — a"b") ch 0" sh 0" + {a' a" + &'&") ch 9 sh 9} 

 + / 2 { (a' 2 ch 2 0" + a" 2 sh 2 0") ch 2 9 

 + 2(a'&' + a"£" ) ch 6" sh 0" ch 9 sh 9 

 + O' 3 sh 2 0" + &" 8 ch 2 6»") sh 2 9} 

 = { (a7/ ch 2 0" + a"/;" sir B") ch 2 9 

 + (a' 2 -f b" + a" 2 + b" 2 ) ch 0" sh 0" ch 9 sh 9 



+ (a'V sh 2 0" + a"&" ch 2 0") sh 2 9} 2 . 



The point ($, -q) is at the same time moving on 

 ( {a' a" + &'&") (ch 2 0" + sh 2 0") (ch 2 9 + sh 2 9) + a'b" + a"&' 

 -f 2 (a' 2 + &' 2 + a" 2 +&" 8 ) ch 0" sh 0" ch 9" sh 9" ) } 2 

 = £ 2 { [ (^ + &"*) (ch 2 0" + sh 2 0") + 2&'&"] ch 2 9 

 + 4(o'&' -f a"b") ch 0" sh 9" ch 9 sh 9 



56 



