OP ARTS AND SCIENCES. 383 



As the sphei-Ical hyperbola is simply made up of the halves of two 

 equal ellipses, it is uot necessary to deal with it separately ; for what- 

 ever is true of the ellipse is true of the hyperbola. 



16. There are certain relations between a, j3, and c, which enable 

 us to reduce some forms of equations. 



cos a 



cos c =^ -, tan a ^=. a, tan h =^ o; 



cos 3 ' r ' 



1 1 



sin^ c b^ 4-1 a-! 4- 1 a^ — lfi 



• • 



(6-'+l)(a-'+l) 

 1 1 



(«--^-)(l + ?'^) _ ,2 



17. By Art. 11,. sin q sin p' = ^•; and from this can be proved a 

 property of the foci similar to one in plane conies. Since the foci of 

 a conic are the poles of the cyclic arcs of the supplementary conic, the 

 distances of the foci from, any tangent are equal to the distances of the 

 corresponding point of the supplementary conic from its cyclic arcs. 

 Hence, if 8 and 5^ denote the distances of the foci from a tangent, 

 sin 8 sin 8^ = constant. 



This constant can be determined as follows : — Using the pole of the 

 tangent as in Art. 10, we have 



a'b'^ + h'x''' 4- aY- = a%%aW — ^—^ x'^ + a^), 



a'^-h'^ ^^(«^+l) 2/70 I ,x aHai-\-\) 

 o — = — ; 1 a (o-' + 1) = ; 



. f. 6(a — ex') . b[a + e.r') 



. . sm _ ^^^^ _ ^,^,,,^^^^ _^ j^^i, sin 0, _ ^^^^ _ ^^^,^^^^^ _^ ^^^|, 



. ,. . o b-i 



Sin sin 0, = . . 



^ a'-i 4- 1 ' 



