9° 



KANSAS UNIVERSITY QUARTERLY. 



The base circle is the locus of the instantaneous centre for all points 

 on the limacon. Let B O P be a line cutting a circle in B and Q. 

 Let the line revolve about B, Q following the circle; the point P will 

 trace a limacon. 



Now, for any instant, the instantaneous center will be the same wheth- 

 er Q be following the circle or the tangent at the point where the line 

 cuts the circle. Therefore the instantaneous center for the point P is 

 found by erecting a perpendicular to the line P B, through B, and a 

 normal to the circle at Q. (Williamson's Diff. Cal. Art. 294). The 

 intersection (C) of these two lines is the instantaneous center for the 

 curve at the point P. But by elementary geometry C is on the circle. 

 Now as the line P B revolves through 360 *> around B, the line B C 

 which is always perpendicular to it also makes a complete revolution 

 and the instantaneous center C moves once round the base circle. 



Below we give a list of theorems obtained by inverting the corre- 

 sponding theorems respecting a conic. In these theorems any circle 

 through the pole is called a nodal circle, any chord through the pole is 

 called a nodal chord, and the line through the pole perpendicular to 

 the axis of the curve is called the latus rectum. The letters e and / 

 signify respectively the eccentricity and half the latus rectum of the in- 

 verted conic. 



The locus of the point of inter- 

 section of two tangents to a para- 

 bola which cut one another at a 

 constant angle is a hyperbola hav- 

 ing the same focus and directrix 

 as the original parabola. 



The sum of the reciprocals of 

 two focal chords of a conic at 

 right angles to each other is con- 

 stant. 



P Q is a chord of a conic which 

 subtends a right angle at the fo- 

 cus. The locus of the pole of 

 P Q and the locus enveloped by 

 P Q are each conies whose latera 

 recta are to that of the orignal 

 conic as ^ 2:1 and i : y 2 re- 

 spectively. 



The locus of the point of inter- 

 section of two nodal tangent cir- 

 cles to a cardioid which cut each 

 other at a constant angle is a lim- 

 acon having the same double point 

 and director circle. 



The sum of any two nodal 

 chords of a limacon at right angles 

 to each other is constant. 



If P and Q be two points on a 

 limacon such that they intercept 

 a right angle at the node, then the 

 locus of the point of intersection 

 of the two nodal circles tangent 

 at P and Q respectively, is a lim- 

 acon whose latus rectum is to that 

 of the original limacon as 

 }4^ 2 : I. And the envelope of 

 the circle described on P Q as a 

 diameter is a limacon, whose lat- 

 us rectum is to that of the original 

 lima9on as i : ^| 2 



