ELLIPTIC AND HYPERBOLIC SECTORS. 439 



and A2 and B, the coefficients z' in the expansions of (m-^)""' and {u-z)'^\ and in 

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

 then it is easy to see that 



C„_, = Ao + A, + A, + A3 + &c. 



Now we have evidently 



Ao=M"-" B„=w"-^ 



A, = -(/»-2)«*"-' B, = -(?i-3)M"-* 



J. . ^ X • ^ 



- 1.2.3 '~ 1.2.3 



&c. &c. 



Observing now that /(««)= X„ = a; Cj„_,) - C(„_2„ 

 and F(wa)=Y„=yC„_„ 

 we have, after replacing uhy 2x, 



M 



2/(. a)=(2.r-.(2 .)-+!^^^(2 .)--!^fc|i!^(2 .)- + &c. 

 F (« «)=j. {(2 xr^-in-2) (2 .)- 4-fcMiz|)(2 ^)^._ (^-4)(>.-5)(n-6) ^^ ^y_, ^^^ 



These expressions for/ [n «) and F (»^ «) are entirely independent of the quan- 



tity c==t= 75 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 formulae 

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

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

 enable him to express them by general formulae, otherwise than by shewing how 

 any number of particular cases might be found. 



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

 we must find a second development of the fraction \_2xz + z' ' 



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

 VOL. XVI. PART II. 3 U 



