4.4-1 PROFESSOR WALLACE ON ANALOGOUS PROPERTIES OF CO-OllDINATES OF 



We have now from Formula} 0, in either curve 

 or, putting t for T («)=-, so that )/=tx, 



{T(«a)}Vc= (i^,^,)„^(i_,^,)„ Q 



In the ellipse, because c is negative, so that -J—c=^ + c.J—l, the formula iur 

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

 sions (1 +' V <•)" and sJO--t^cy are expanded into series; and united by sub- 

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

 cients of ^, f, t^, t^, &c. in the expansion of the binormal (1 + 0", 



rp ^_ Ai/ + A,c P + AjjC^/^ + &c. p 



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 hyperbola, 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 -/c and — for t, we have then in the hy- 

 perbola, 



'Tjna)_ \a bf \a b\ ^ 



"^ ~ f^^iv^ j£_^i" 



U 6J \a bf 

 X y x^ v^ x^ v^ ix v\ ix v\ 



Let us now put -+\=r, then, because — -^=l ; and —-^—= K-^-A \---t\ ■ 



^ a b a' b' a' o' \a b) \a b) 



therefore, ;• (--f) =1. and --|=-: Formula S may now be expressed thus, 



T (>»«) _ r^y-\ ^ , 6+T(>»«) 



/•"-f — 



and putting - for 2 n, and ^^ for w ; 



