ELLIPTIC AND HYPERBOLIC SECTORS. 



43.9 



and A. and B, the coefficients z' in the expansions of (m-^)""' and («-?)""', and in 

 general A, and B^ the coefficients z" in the expansions of {u-z)""''' and («-«)""""'■, 

 tlien it is easy to see that 



C„_, = A„ + A, + As + Aj + &c. 

 C„_, = B„ + B, +B, + B, + &c. 



Now we have evidently 



Ao=m"-' 



A, = -(>i-2)m"^ 



(w-3) (w-4) „_5 

 ^ 12 



^'- 1.2.3 



Sec. 



B„ = m"-- 



^'~+ 1.2 " 



R- (>'-5)(>>.-6)(».-7) 

 •1.2.3 

 &c. 



2 C „_, = 2 .--2 (»-3) ."-' + 2(.-4)(.-5 ) ^„__ 2(.-5) (.-6) (.-7) ,^^„_ ^ ^^_ 



Observing now that f(n a)=X„=x C(„_„-C,„_2„ 

 and F(«ai)=Y„=yC„_„ 

 we have, after replacing uhy2a:, 



M 



1 . 2 



1 . 2 



F in a)=, {(2 .)"-' -(.-2) (2 ,r^ + (^L:^i!^^2 ,)^._ (^-4) (.-5) (.-6) ^^ ^^„_, ^^^ 



These expressions for/ (?? «) and F (w «) axe entirely independent of the quan- 



tity c==t ^ they are, therefore, identically the same for the ellipse and hyperbola. 



And if the axes of the ellipse be supposed equal, they become the known formulse 

 for the cosine and sine of the multiple of an arc, which, in substance, were found 

 by ViETA.* The angular analysis was not, however, sufficiently advanced to 

 enable him to express them by general formulae, othei-wise than by shewing how 

 any number of particular cases might be found. 



11. That we may obtain values of the functions /(« a), F (n «) in another form, 

 we must find a second development of the fi-action i_2 3-!: + z' ' 



* Francisci Vieta Opera Mathematica. Leyden, 1646 (pp. 295, 297). 

 VOL. XVI. PART n. 3 u 



