92 



REPORT — 1856. 



The point T<r on the circle is the pole of the tangent PT to the parabola, 

 and the point P on the parabola is the pole of the tangent -mr to the circle. 



As the parabolic arc VP— PT is the logarithm of the number VN, so the 

 circular arc Atzr is the logarithm of the bent line Atsr+nrS. 



The locus of the point r, the foot of the perpendicular from S on the tan- 

 gent to the circle at cr, is a cardioide whose cusp is at S, and whose diameter 

 is that of the circle. 



While the circle is the 2)olar reciprocal of the parabola, the cardioide is its 

 inverse curve ; for the cusp polar equation of the cardioide is |0=2r(l + cos0), 



2w2 

 while the focal equation of the parabola is p^= -j— ; -; hence p|0^=4m^ 



Since the parabola and the circle are reciprocal polars one of the other, the 

 circumference of the circle passing through the focus of the parabola, we 

 have been able by the help of this reciprocal circle to give geometrical repre- 

 sentations, as in XII. and XIV., of the properties of the trigonometrj' of the 

 parabola. 



There is this further analogy between the properties of the circle and those 

 of the i^arabola, — that as the arc which is equal to the radius subtends no 

 exact submultiple of any number of right angles, however large, so in the 

 parabola the angle or amplitude which gives the tangential difference or 

 logarithm equal to the modulus is incommensurable with any number of right 

 angles. In the former there are 206265 seconds, in the latter there are 

 178575 seconds*. 



The theorem given above, that a parabola is the reciprocal polar of a circle 

 whose circumference passes through its focus, suggests a transformation 

 which will exhibit a much closer analogy between the formulae for the recti- 

 fication of the parabola and the circle, than when the centre of the latter 

 curve is taken as the origin. 



XXXIII. Let SB A be a semicircle ; let the origin be placed at S ; let the angle 



* It is worthy of investigation to ascertain whether any relation can be found between 

 the angle or arc (1), and the angle e which gives the tangential difference equal to the mo- 

 dulus in the parabola. 



