444 PROFESSOR WALLACE ON ANALOGOUS PROPERTIES OF CO-ORDINATES OF 



We have now from Formulae 0, in either cm^ve 



Tr \-L (^+.yVg)"-(^-yVg)" 



or, putting t for T (a)=-, so that y=tx. 



In the ellipse, because c is negative, so that V-c=v' + e.V— 1, the formula in- 

 volves the imaginary symbol V-1: this, however, disappears when the expres- 

 sions (1 + ^ V c)" and s/(l-ts/ c)" are expanded into series ; and united by sub- 

 traction and addition : We have then, putting Aj, A^, A,, A^, &c. for the coeffi- 

 cients of t, t^, f, f, &;c. in the expansion of the binormal C^ + fT, 



rp, ^_ Ai<' + AgC (^ + Aggg/^ + Sic. „ 



For the circle or ellipse, the terms in this formula containing the odd powers of c, 

 viz. the first, third, &c. must have the sign — , and the remainder the sign + : But 

 in the h3rperbola, they must all have the sign + ; in either case the expression 

 for the tangent of the sector contains only real quantities. 



15. In formula Q let us put ~ for Vc and — for t, we have then in the hy- 



X 



perbola, 



XV X^ V^ X^ V^ i X v\ / X v\ 



Let us now put -+\=r, then, because ^-7^=1 ; and -i--rz= (- + 7) ( — t) ; 



^ a b a^ b^ a* 0* \a b) \a 0/ 



therefore, r (-— f) =1> ai^d --|=-: Formula S may now be expressed thus, 



T(na) _ ^_r--l , 6 + T(n«) 



and p-utting - for 2 n, and ^^ for w ; 



\2m/ 



r = 



V2 m/ 



