( «51 ) 



Fiö.3 



3. Application to the cardioid inotion. Let A(J be the right line 



/, wliich passing tlirougii tiie fixed 

 point C of circle {0) glides with one 

 of its points A along the circumfer- 

 ence ; let now too the demand be to 

 construct the conic A^ corresponding 

 to / (tig. 3). 



Circle ((>) is the cuspidal circle; 

 the pole F lies diametrically opposite 

 to A and the inflectional circle (J/) 

 is symmetric to {0) with respect to 

 the tangent in P. Now ).\ too, is to 

 be constructed according to the preced- 

 ing point by point; this takes place 

 in the following way : 



Let D be a point of /, draw DF 

 and DI; the normal in F on BF intersects DI in Q, the parallel 

 to Fl out of Q intersects DF in (f. 



Just as with the elliptic motion we can again construct some par- 

 ticular points. If we aj)pl3' the general construction to point 6', it is 

 evident that y lies halfway CF; is evidently a point of A" and [3 

 is the centre of curvature of the point at infinity on /; so the conic 

 P." passes through y, O, ■?, and osculates circle (0) in F. 



Whilst thus the construction of P.' offers no difficulties, the gene- 

 ra'ed system of conies is more intricate than the preceding. 



Some properties are to be found geometrically ; thus it is soon 

 evident tiiat the system contains two parabolae. 



For a parabola is necessary that ylC be a tangent to the inflectional 

 circle (3/). Let us imagine the two touching circles {()) and (M) 

 and if we draw from the endpoint A of the common diameter the 

 tangents to circle (M), we see that we can give two positions to 

 (J/) so that one of the tangents passes through (\ so there are two 

 j»ai'al)olae. belonging to the system. From the figure is evident: 



1 

 sin rr = — . 

 ' 3 



From this ensues : For all values of / ACQ, which are lying between 



1 

 the values (p z= sm~^ — on one side as well as on the other of CO. 



3 



/- becomes a hyperbola, for all values outside those limits X- be- 

 comes an ellipse, the transition between the ellipses and the hyper- 

 bolae is formed by two parabolae. 



The locus of the centres of this system 'of conies does not become 



