ELLIPTIC AND HYPERBOLIC SECTORS. 437 



Three of these formulae, viz. the first, third, and fourth are exactly the same 

 for the ellipse and hyperbola ; in the remaining one (the second), the sign of the 

 term that contains F {n «) changes, it being negative for the ellipse and positive 

 for the hyperbola. 



8. From these general formulae, we may deduce any number of particular 

 values, by making n equal to 0, 1, 2, 3, &c. successively. Thus, from the first 

 and third, we find 



/(0«)=1 



F(0«)=0 



/ («) =^ 



F («) =y 



/(2«) = 2:ir^-l 



F(2«) = 2;^^ 



/(3a) = 4x'-3x 



F(3«) = (4ar'-1)^ 



/(4«) = 8a;^-8a;- + l 



F(4.a)=(8x'-4:x)^ 



/(^5a) = 16:t^-20x' + 5x. 



F(5a)=(16x'-12x' + l)t/. 



&c. 



&c. 



and in this way we may proceed to any extent in deducing formulae, which are 

 identically the same for the ellipse and hyperbola. 



9. Since the relation of every two adjoining terms of the two series/ (a), 

 /(2a), (/3a), kc, and F (a), F (2a), F (3«), &c. is exactly the same in the two 



curves it follows, that n being any whole number, / {n «), and F {n «) wiU be the 

 same function of x and p in the two curves ; this was the property particularly 

 noticed by Maclaurin, in the case of /(wa),* and employed in deducing Vieta's 

 Theorems from a property of an equilateral hyperbola derived from the theory of 

 logarithms. I shall now deduce the same properties from the formulae which 

 have been here investigated. 



10. To abridge, let us denote/ {n a) and (F n «), the co-ordinates of an elliptic 

 or hyperbolic sector (which is equal to the sector denoted by « taken n times), by 

 the more simple symbols X„ and Y„, then, a;, and j/ as before, being put for the co- 

 ordinates of the sector a, the general formulae 



/{(n-l)a}+/{ (n + l)a] = 2xfina), 

 F{(n-l)a} + F{(n + l)a] = 2xF(na), 



will be expressed by fewer characters, thus, 



^ n-l + X „ + , = 2 X X„, 



Y^, + Y„^, = 2;.Y„. 



Let z denote an arbitrary quantity, which is to be introduced as an analytical 

 artifice to generate a series of terms ; and, by giving successive values to the num- 



* Maclaurin's Fluxions, Article 757. 



