441 



ELLIPTIC AND HYPERBOLIC SECTORS. 



From these formulse, puttmg/(a) for x, and F (a) for y, we have 



/(w a) + F (» «) Vc = {/(a) + F (a)Ve}", 

 /(»»«)- F(»»«).Vc={/(«)-F(a)Vcr; 

 and hence, again, n and m being any whole numbers, 



I = //(« a) + F (re a) . ^c | "=", 



/(«) + F(«)Vc( 



= |/(M«) + F(raa).s'e]-"; 



therefore, /(»? «) + F (w a) . \/c = | /(» «) + F(w «) . Vc } » ; 



and, putting na=a', so that ma— — a'; we have 



f (?"') + ^ (^'"') -^"^ {/(a') + F(«'). V4" = 



and putting « instead of a, and again x and ?/ instead of/(a) and F («), 



Exactly in the same way we prove that 



In the hyperbola x'^—cy-={x+y Vc) {x—y \/c)=l, 



and in general {/(»a) + F (rea). a/c] {/(wo)— F (»«) . -s/c) } = 1. 



Now, as in the circle, we consider the cosine of a positive and negative angle 

 at the centre to be equal in magnitude, and to have the same sign ; but their sines 

 to be equal in magnitude, with contrary signs, so, by analogy, in the ellipse and 

 hyperbola, we must reckon f{ + na)-f{-na), but F ( + w «) = _ p ( _ » „) ; and hence 

 we have 



{/(wa) + F( + rea) Vc} {/{-n a) + ¥ {-n a) . V c} = \, 

 and/(-»a) + F(-wa).V'c ={/(+»a) + F( + wa).\/c}-' = (a;+y\/c)— . 



In the same way we find/(-»8 «)-F (— » «) . Vc=(jic—y Vc)-". 



So that, on the whole, whether w be a positive, or a negative whole number, or a 

 fraction ; in each case 



/(« a) + ¥ (n a) . ^/ c ={x +y \/ c)" ; 

 ./(» a) — F{n a,).Vc = (x—y Vc )" . 



::} 



N' 



