3i.] PLANE MOTION. I7 



the perpendicular to the plane of motion of any point through 

 the centre of motion of this point ; and that the centrodes are 

 now not curves, but cylindrical surfaces rolling one upon the 

 other. 



31. Exercises. 



(1) Show how to find the direction of motion of any point /'rigidly 

 connected with the connecting rod of a steam engine. 



(2) A wheel rolls on a straight track; find the direction of motion 

 of any point on its rim. What are the centrodes in this case ? 



(3) Show how to construct the normal at any point of a conchoid. 



(4) Find the equation of the fixed centrode when a line V of a 

 plane figure always touches a fixed circle O, while a point A of V moves 

 along a fixed line /. 



(5) Show that, in (4), the fixed centrode is a parabola when the 

 fixed circle touches the fixed line. 



(6) Two straight lines / f , /" of a plane figure constantly pass each 

 through a fixed point O', O" ; investigate the motion. 



(7) Four straight rods are jointed so as to form a plane quadrilateral 

 ABDE with invariable sides and variable angles. One side AB being 

 fixed, investigate the motion of the opposite side ; construct the cen- 

 trodes graphically. 



(8) Let a straight line / in a fixed plane be brought by a finite 

 displacement from an initial position / into a final position 4 ; and let 

 P be any point of/, P Q its initial position (in / ), P l its final position 

 (in 4) . Then the following propositions can be proved : 



(a) The middle points of the displacements P^ of all points P of 

 / lie in a straight line ; 



() the lines /o/i envelop a parabola ; 



(<:) the projections of the displacements P^Pi on the line joining 

 their middle points are all equal ; 



(d) if / have a continuous motion in the plane, the tangents to the 

 paths of all its points envelop a parabola of which the instantaneous 

 centre is the focus and / the tangent at the vertex. 



PART I 2 





