12 



GEOMETRY OF MOTION. 



[24- 



The condition that a curve of the moving figure should always 

 pass through a fixed point may be regarded as a special case of 

 the condition just mentioned, one of the fixed curves being 

 reduced to a point. 



24. Any curve of the moving figure forms during the motion 

 an envelope, the points of the envelope being the intersections 

 of the .successive infinitely near positions of the moving curve. 

 Let /, /' be two such successive positions of the curve, A their 

 intersection, C the instantaneous centre ; then CA is perpen- 

 dicular to / as well as to /', and hence to the envelope. The 

 envelope can therefore be constructed by letting fall normals 

 from the instantaneous centres on the corresponding positions 

 of the generating curve. 



25. The following examples will illustrate the method of 

 finding the centrodes and the path of any point of the moving 

 figure in plane motion. 



Elliptic motion : Two points of a plane figure move along two 

 fixed lines that are at right angles to each other. 



Let A y B (Fig. 6) be the points moving on the lines Ox, Oy ; 

 the perpendiculars to these lines erected at A and B intersect 

 at the instantaneous centre C. Denoting by 2 a the invariable 



distance of A and B, we have 

 OCAB=2a for all posi- 

 tions of the moving figure. 

 The fixed centrode (c) is 

 therefore a circle of radius 

 2 a described about the in- 

 tersection O of the fixed 



lines. 



Fig. 6. 



To find the body centrode 

 (c 1 ) we must construct the 

 triangle ABC for all possible 

 positions of AB. As BCA is always a right angle, the body 

 centrode will be a circle described on AB as diameter. Hence 



