ANALYTIC GEOMETRY 



[Cn. Mil. 



190. The Neilian, or semi-cubical, parabola.* This curve 

 may be <lt -t HTAXKL be a given pai -al>- 



ola whose equation is 



y a = 4jw:; . . . (1) 



let TMS be any double ordinate of 



the curve, T7\ a tangent at the point 



Tm(x v y^*atiLAQ& perpendicular 

 from the vertex upon this tangent ; 

 if QA intersects TS in P, then the 

 locus of P as T moves along the 

 parabola is called a semi-cubical or 

 Neilian parabola. 

 Its rectangular equation is derived as follows : the equa- 

 tion of TTi is 



*- 



hence the equation of AQ is 



*=-ft< 



The equation of TS is 



*-*,. . . (4) 



If now equations (3) and (4) be regarded as simultaneous, 

 then x and y are the coordinates of the point P in which the 

 two lines intersect, and if x l and y l be eliminated by means 

 of the equation 



an equation connecting x and y is obtained. 



This curve is historically interesting, because it is the first one winch 

 was rectiflfd, <.., it is the first one the length of an arc of which was 

 expressed in rectilinear units. This celebrated rectification was perform, .j. 

 without the aid of the modern Calculus methods, by William Neil, a pupil of 

 Wallis (see Cantor, Geschichte der Mathematik, IM. II., S. 827), in 1667 ; tin- 

 is therefore called the Neilian parabola. It is also called the . 

 cubical parabola because its equation may be written in the form y = ax%. 



