436 PROFESSOR WALLACE ON ANALOGOUS PROPERTIES OF CO-ORDINATES OF 



ceeding from these formulae, we might estabhsh a complete theory of the conic 

 sections. I shall, however, only farther extend the investigation to the co-ordi- 

 nates of sectors, which are multiples of a given sector of an eUipse or hyperbola, 

 and are contained between the transverse axis and any semidiameter. From 

 these the whole theory of angular sections may be deduced, also the correspond- 

 ing properties in an equilateral hyperbola. 



7. Let a and h be the axes of a conic section, a being the transverse, and h the 

 conjugate, or, in the case of the hyperbola, the second axis. This assumption will 

 require no change in the formulae. We may introduce e, the eccentricity of the 

 curves instead of the second axis 5, and since in the ellipse V=(^—i, and in the 

 hyperbola V-i-a^; if in either curve «, |8, «-|-/3, a -j8 denote sectors contained 

 between a, the semitransverse, and semidiameters, the co-ordinates of whose ver- 

 tices (which may be called the co-ordinates of the sectors) are respectively, 



/(«). /(^), /(« + /3), /(a-/3), 



F(«), F(/3), F(« + /3), F(«-/3), 



In either curve, 



a. /(« + /?)=/ («)./(/S)- 



a' 



a—e- 



,F(«).F(/3), 



(1) 



a.F(« + /3) = F(a)./(/3)+/(a).F(/3), (2) 



a./(«-^)=/(a)./(/3) + 



a 



a' — e' 



a.F(«-/3) = F(«) .fip)-f{«) . F (/3). 



F («) . F (/3), 



(3) 

 (4) 



G. 



In these formulae no attention to the signs of the terms is now required, be- 

 cause they are adapted to each curve, by the circumstance of the eccentricity of the 

 elhpse being less, and that of the hyperbola greater than the transverse axis. 



By adding and substracting (1) and (3), also (2) and (4), and, to simplify, 

 making the semitransverse a=l, we obtain 



/ (« + /3) +/ (« - ^) = 2/(a)/(/3), 



2 



/(« + ^)-/(«-/3) = 



F(a)FO), 



(5) 

 (6) 



H. 



/ 



F(a + /3) + F(«-/3) = 2F(«)/(/3), (7) 



F(« + ^)-F(a-/3)=2/(«)F(/3) (8) 



By putting n « instead of «, and a instead of /3, also x for / («), and y for 

 F (a), these formulae become 



f{(^n^l)a}+f{{n-\)'a} = 2x.f{na)- 1 



2 



f{{n + l)a}-f{{n-l)a} = :^—^y.¥{na); 2 



¥{in + l)a{ + ¥{{n-V)a} = 2x.¥ {na); ..... 3 

 ¥{(n + l)a}-¥{{n-l)a}=2y.f(na) 4 



( 



K. 



