( 648 ) 



Mathematics. — ''Some constructions deduced from the motion oj 

 a plane system." By Prof. J. Cardinaal. 



(Communicated in the meeting of February 26, 1908.) 



1. We precede two wellknown principles of motion. 

 a. Let the motion be generated by rolling a curve 6» (body centrode) 

 with which the system iS'is connected over a curve CJ (space centrode). 



If A, B, C, are moving points of >S' and «, /?, y . . . . the centres 



of curvature of the orbits which they describe at a certain moment 

 there exists between the system S and the system ff, f?, y . . . . (^) a 

 quadratic correspondence and such that if «, /?, y . . . were moving 

 points A, B, C . . . would be the centres of curvature of their orbits. 

 The conies of S corresponding to the right lines of ^ touch the 

 tangent of body centrode and space centrode in the pole and osculate 

 each other. The inflectional circle belongs to it. The reverse theorem 

 is easy to deduce. 



b. Let P be the pole (fig. 1), 

 / the inflectional pole (common 

 point of the tangents in the mtlec- 

 tional i)oints) ; so the inflectional 

 circle is known. Let A be a 

 moving point, then « is determined 

 as follows : Draw AI and AP; 

 determine the point of intersection 

 Q of AI with the normal through 

 P on A P. Draw out of Ö the 

 parallel to IP, which cuts Pin «. 

 2. Application to the elliptic 

 motion Jig. 2). Let AB (I) be the 

 right line gliding with its points 

 A and B along the rectangular 

 axes IX and IT and let the 

 demand be to construct the conic X^ 

 cori-esponding to /. 

 The ^'ircumscribed circle (J/) of A ABI is the inflectional circle ; 

 P is also directly known ; the centre of curvature belonging to a 

 |)oiMt of AB can be constructed according to (16) and so each 

 point of /' can be determined. However, some points of X' are 

 immediately known. The centre M of AB describes a circle having 

 I as cenrre, the centres of curvature a and /? belonging to A and B 

 lie at inlinitt; distance in (lie directions IB and lA, /- furthermore 



