i6 



GEOMETRY OF MOTION. 



[29. 



29. Conchoidal Motion : A point A of the figure moves along a 

 fixed straight line 1, while a line of the figure, 1', containing the 



point A always passes 

 through a fixed point B 

 (Fig. 10). 



The fixed point B may 

 be regarded as a circle 

 of infinitely small radius, 

 which the line /' is to 

 touch. The instantane- 

 ous centre is therefore 

 the intersection C of the 

 perpendiculars erected at A on / and at B on I'. 



The fixed centrode is a parabola whose vertex is B. To 

 prove this we take the fixed line / as axis of y, the perpendicular 

 OB to it drawn through the fixed point B as axis of x. Then, 

 putting ^.OBA=(j> and OB = a, we have for the co-ordinates 

 ofC 



10. 



ya tan< ; 



hence xa=y 2 /a, or, for B as origin and parallel axes, y*=ax. 



The equation of the body centrode, for OB, OA as axes of x 

 and;j/, is a\x 2 +y*)=x*, or r cos 2 Q a. 



The points of /' can easily be shown to describe conchoids, 

 whence the name of this form of plane motion. 



30. The results obtained in the preceding articles for the 

 motion of a plane figure in its plane apply directly to the motion 

 of a rigid body, if any one point of the body describes a plane 

 curve while a line of the body remains parallel to itself. For in 

 this case all points of the body move in parallel planes, and the 

 motion in any one of these planes determines the motion of the 

 whole figure. 



The only modifications required would be that instead of an 

 instantaneous centre we should have an instantaneous axis, viz. : 



