261 



circle from the focus of the parabola. This follows from an inspec- 

 tion of the figure ; but it may easily be proved as an independent 

 theorem. 



XXIII. The vertical tangent OT bisects all the cords of the Lo- 

 gocyclic passing through F. The angles VRT or VR,T=i// and 

 6 or OFR are so connected that tan 4/=cos 0. Hence the maximum 

 ordinate of the loop is found by making t//=0, or tan 0= cos 0, or 



XXIV. If any point Q on the parabola be taken as centre, and 

 through the two corresponding reciprocal points on the Logocyclic 

 curve a circle be drawn, it will always cut at right angles the fixed 

 circle whose centre is the focus F and radius ==a. 



XXV. The Logocyclic curve, like the circle, is its t)wn inverse 

 curve. 



On the Area and rectification of the Logocyclic Curve. 



XXVI. The area of the loop will be found equal to 



and the area between the asymptote and infinite branch will be 





Hence the entire area of the curve is 4 a 2 , or equal to the square of 

 the semiparameter of the parabola. It is obvious that the area of 

 the half-loop is equal to the difference between the square of a and 

 the quadrant of a circle inscribed in it, while that of the infinite 

 branch is equal to the square of a with the quadrant added to it. 

 Hence also the difference of the two areas is equal to aV, that is to 

 the area of a circle whose radius is a. 



The length of the half-loop, together with that of the infinite 

 branch of the curve, is equal to the infinite branch of an equilateral 

 hyperbola whose transverse semiaxis is V 2#, and to the half-loop of 

 the lemniscate which belongs to that equilateral hyperbola. Hence 

 the entire length of the Logocyclic is equal to the entire continuous 

 arc of an equilateral hyperbola whose transverse axis is \/2a, and 

 to the loop of the lemniscate which belongs to that branch ; while 

 the difference between the lengths of the loop and the infinite 

 branch is equal to an arc of the parabola together with a right line. 



