ON THE TRIGONOMETRY OF THE PARABOLA. 93 



ASB=^; and let D, as before, be the diameter of the circle. Through B 

 draw the tangent BP ; let fall on this tangent the perpendicular SP=p, and 

 let BP, the subtangent, be equal to t. 



Now asj9=Dcos"S, and <=Dsin ^cosS^, as also the angle ASP=2S-, if 

 we apply to the circle the formula for rectification in IV., we shall have 

 the arc 



AB=s=2DJcos2^6?^— Dsin^cosS- (38) 



The subtangent to the circle, which is exhibited in this formula, disappears 

 in the actual process of integration ; while in the parabola, the subtangent 

 which is involved in the differential is evolved by the process of integration. 



As in the parabola, the perpendicular from the focus on the tangent bisects 

 the angle between the radius vector and the axis of the curve ; so in the 

 circle, the radius vector SB drawn from the extremity of the diameter, bisects 

 the angle between the perpendicular SP and the diameter SA. 



It is easily seen that while the line SB makes the angle with the axis, the 

 line SP makes the angle 26, and the perpendicular SR on the tangent to the 

 cardioide makes the angle 30 with the axis. 



Hence if we take the reciprocal polar of the cardioide, the line drawn per- 

 pendicular to the tangent at any point on the curve trisects the angle between 

 the axis and this radius vector. Consequently the polar reciprocal of the 

 cardioide is a curve, such that if a point be taken anywhere on the curve, 

 and a perpendicular be drawn to the tangent at this point, it will trisect 

 the angle between the axis and the radius vector drawn to the point of con- 

 tact. Hence the reciprocal polar of the cardioide enables us to trisect an 

 angle, in the same way as a parabola gives us the means to bisect it. 



XXXIV. To determine the tangential equation* of the reciprocal polar of the 

 cardioide. The radius vector u of the cardioide being connected with the 

 polar angle 6 by the equation u=r(l + cos 6), and p being the perpendicular 



on the tangent of its polar reciprocal, we shall have — = — (I + cos 6). 



Let jO= — , then as cos d=p^ and — = -/IMV, ^ and v being the tangential 

 coordinates of the curve, we shall have 



Consequently l(p^y^)-p^y--p^(^+y^)=o (39) 



is the tangential equation of the reciprocal polar of the cardioide. The 

 common equation of the cardioide, the cusp being the pole, is 



l(^+f)-rxy-f^(x'+f)=0. ..... '(40) 



The reader will observe, that the equation between the coordinates x and 

 y of the cardioide is exactly the same as the equation between the tangential 

 coordinates E and v of the reciprocal polar of the cardioide. 



XXXV. The quadrature of the hyperbola depends on the rectification of 

 the parabola. 



Through a point P on the parabola draw a line PQ parallel to the axis 

 and terminated in the Tertical tangent to the parabola at R. Take the line 

 RQ always equal to the normal at P, the locus of Q is an equilateral hyper- 

 bola. For x=2m sec <^, and as before y=2m tan 0, therefore 



x^—f=im?, . (41) 



* Tangential coordinates, p. 70. 



