ELLIPTIC AND HYPERBOLIC SECTORS. 441 



From these formulae, putting/(a) for x, and F («) for ^/, we have 



f{n a) + F(n «) .>Jc = {/(«) + F (a). V^}", 

 /(wa)-F(«a)Vc={/(«)-F(«).Vc}"; 



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



[=|/(«a) + F(/^a).VcU, 

 /(a)+F(a)V^<^ ,11 



I = -! /(m a) + F (m «) . s'c > -» ; 



therefore, f{m a) + F (m «) . \/c = | /(% a) + F(w «) . Vc [ « ; 



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



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



Exactly in the same way we prove that 



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

 and in general {/(w a) + F (« «) . V^c } {/(w «)— F (w a) . \/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 

 h3rperbola, we must reckon /( + »«) =/( - n a), but F( + ?i«)= _F(-wa); and hence 

 we have 



{/(w a) + F (+««).'/<;} {/(-wa) + F(-wa) Vc} = l, 

 and/(-«a) + F(-wa).\/c ={/( + % a) + F(+w«).Vc}-'=:(x+^\/c)-". 



In the same way we find / ( — w «) — F ( — w «) . a/c =(« — y v'c)-". 



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

 fraction ; in each case 



/(«a) + F (na) .\/c=(ar+y a/c)"; \ 



fin a)- F{na).Vc = {x-y 's/c )" 



:} 



