51 



Ellcry W . Davis 



The black one is 



.r', />' cos & sh <f> + a" sin 0' ch 



y', — &" cos 0' ch cf> -f- a' sin 6' sh <£ 



a' cos 6' ch 4> + &" sin B' sh <£, .r' 



- a" cos 0' sh <f> + &' sin 0' ch <£, y' 



a'cos 0' ch <£ 4- & " sin 0' sh <£. &' cos 8' sh + a " sin 0' sh </> 



— a" cos 0' sh <f> + &' sin d' ch c/>, — &" cos 0' ch <£ + a' sin 0' sh </> 



or, sav, 



(D.i-' — Cv')' J — (— Bx' + Jv') 2 = (AD — BC)-. 



The bine one can be similarly written down from the equa- 

 tions for $ and ?/. Or what is the same thing, in the A, B, C, D 

 for a', a", b', b" substitute respectively, a' -f- a", a" — a', b'-\-b", 

 b" — V and for (.r', /), (|, v ). 



Somewhat less interesting' are the non-central conies unless we 

 except those in which the terms of the second degree are the 

 square of a circular ray. For then, just as the square equated 

 to a constant represents two lines making any angle you please 

 with each other ; so, when the same square is equated to a linear 

 expression the resulting locus may be regarded either as a para- 

 bola or as an ellipse whose imaginary asymptotes make any real 

 angle you please with each other. 



58 



