48 



INTRODUCTION. 



It has been shown that ABq : 

 CDq::AE:EF (199), therefore 

 ABq : AEq: *. CDq : AE.EF ; but 



^^ the chord of the circle of equal 



^^ . 2 AE.EF , 

 curvature, EG, is = — — -- — , and 

 OH 



AE.EFz:|EG.CH, therefore ABq : AEq: :CDq : I EG.CH: : 2^ 



:EG. 



Scholium. It may easily be demonstrated that a perpendicular 

 to the normal of the curve, or to the line perpendicular to its tan- 

 gent, passing through the point vi^here it meets tlie axis, bisects the 

 focal chord of curvature, and that a perpendicular, falling from the 

 same point on the chord, cuts off a constant portion from it, equal to 

 tlie third proportional to the semiaxes. 



202. Theorem. The square of any ordinate of an 

 ellipsis, parallel to the lesser axis, is to the rectangle con- 

 tained by the segments of the greater axis, as the square 

 of the lesser axis to the square of the greater. 



On the centre A describe the 

 circle BCDE, passing through 

 the focus B; then EFlBF:: 

 CF : DF (138). Call HI,«, 

 HB,5, AB,a:, GH,z, then EF=2ff, 

 BF = 2ft, CF=^BH — 2BG= 

 2GH=z22r, DFzzEF— EDir2a— 

 2x, and 2a : '2b'. \2z '. 2a — 2x, a '. h'. \z '. a — x, a ', «+&: \z '. z-{-a — x 

 : '.a-\-z l 2a — X +6+z (32); also a ', a — b'. \z \ z — {a — x)'. \a — z \ 2a 

 — X — (6+z), and by multiplying the terms, aa '. aa — bb:'.{a-{-z). 

 (a— z) : (2a— x)2— (6+2)2, or Hlq.HKq: IIG.GL ; AFq— GFq, or 

 AGq. 



203. Theorem. The area of an ellipsis is to that of 

 its circumscribing circle, as the lesser axis to the greater. 



For since the square of the ordinate is to the rectangle contained 

 by the segments of the axis, or to the square of the corresponding 

 ordinate of the circle (137), as the square of the lesser axis to that of 

 the greater, the ordinate itself is to that of the circle in the constant 



