ON THE TRIGONOMETRY OF THE PARA.BOLA. 95 



Hence, subtracting the former from the latter, 



U(m . 20) — 2n(m .^)=m tan 20 (sec 29 — 1 ). 



Accordingly, 



the hyperbolic area = mU.(m . 20) — m^ tan 20 (sec 20 — 1 ). . (45) 

 Since 



2 tan 20 =2 tan (p sec ^, 

 we have , 



20=0-L^ '. (4.6) 



Hence the normal angles and of the corresponding points of the para- 

 bola and hyperbola are so related that 



20=0-J-^, 



whence we might at once have inferred the relation established in (44), 

 namely 



(sec + tan 0)2= sec 29+ tan 20. 



The points P and Q on the parabola and hyperbola respectively may be 

 called conjugate points. They are always found in a line parallel to the 

 axis. 



If through the points P and Q on the parabola and hyperbola we draw 

 diameters to these cu7-ves, they will make angles with the normals to them at 

 these points, one of which is the duplicate of the other. 



For these angles are 20 and ^ respectively, 



but 2d=(p-^<l>. 



XXXVI. Let P„, Pj, P^, Pg, P4 . . . P«-i, P„ be perpendiculars let fall 

 from the focus on the n sides of a polygon circumscribing a parabola, and 

 making with the axis the angles 0, 0, 0-L0, 0-^0 -J- 0, 0-L 0-1- 0-1- 9, ... to 

 n terms respectively. 



Let 



(47) 



(48) 



