JH .1 WALYTR QMOMMTB7 [Cit. \. 



two equations [60] are equations <>i the ellipse in terms of 

 the eccentric anijlr, for tnirether they e\piv>s tin- condition 

 that tin- point /' is mi the ellipse. (1).* 



Since, in the figure, A OM'H an 1 OMQ are similar, it 



follows that 



MP:MQ = OR: OQ = b:a, 



and OM'i OM 



that is, the ordinate of any point on the ellipse is to the o ///- 

 nate of the corresponding point on the major auxiliary circl< in 

 the ratio (b-.d) of the semi-axes. Similarly for the abscissas 

 of the corresponding points R and P. 



147. The subtangent and subnormal. Construction of tan- 

 gent and normal. 



L,t g + g-1 . . . (1) 



be a given ellipse, 

 then + M = 1, ... (2) 



is the tangent to it at a point P 1 =(a; r y x ). Let this tangent 

 cut the z-axis at the point T. Draw the ordinate MP r 

 Then the subtangent is, by definition, TM\ and its numer- 



ical value is 



MT=OT- 



2 



but, from equation (2), OT= \ and OM=x^\ 



x \ 



hence 



.e., 



The equations [60] are, of great service in studying the ellipse by the 

 methods of the differential calculus. 



