14 



RESEARCHES ON II. 



But from the theory of the ellipse, it is known that the perpendicular drawn from 

 the centre to the tangent, and the normal N are the extremes of a geometrical 

 proportion, of which the semi-conjugate axis is the mean; that is 



Likewise, the distance of the tangent from the centre measured on the axis of the 

 abscissus, is 



0T = -, 



X 



a being the semi-transverse axis of the ellipse : therefore, substituting these values 

 in (1), and making BO = m, BA = n, we shall have 





making also 



a^ — V^ eV, 



and introducing the value of the normal in the ellipse, 



j2 ha 



N^ = — (d? — e^ x^) , whence — = = — — . 



the value of A? becomes 



db h 



A^ ^n^ + { — ==^=z m — ______ . - ). 



(2) f^l^ia^-x-^), 



Taking now the equation of the ellipse 



and supposing 



(3) X = a sin. ^, we shall have y ^h cos. ^, 

 and ^ will be the amplitude of the ellipse, and is reckoned on the circumference of 

 a circle^ circumscribed round the ellipse, beginning from the diameter of the circle 

 corresponding to the conjugate axis, and terminated where the ordinate of the 

 ellipse produced meets the circumference. 



If 0, PI. I. Fig. 5, be the centre of the ellipse, and ANB' ... the circle which 

 circumscribes it, Xthe ordinate of the ellipse produced to N, 5Wwill be the arc <^, 

 and BON the corresponding angle, and 0X= a sin. ^. 

 Hence, substituting for x its value 



A= b4-( < A-m_bsm^^ \l 

 S Wa^ — a^e'sin.''^' 



by the substitutions (3) we have also 



ds = */ (J? — ^ (^ dn'} ^ . d^; 

 therefore 



(4) Ads = Vr^ (a^ — eV sin? ^) + {db — mb sin. ^f . d ^. 



' Legendre, i, page 15. 



