26;.] 



PLANE MOTION. 



12, 23 (produced if necessary) in points A, A' whose paths 

 are homothetic (similar and similarly situated) curves. For 

 the points 4, A, A' remain always in 

 line and the ratio 4 A/4.A f remains 

 constant. Hence if a pencil be at- 

 tached to A f and A be made to trace 

 a given curve, A' will trace a similar 

 curve. 



Instead of fixing 4, the point A r 

 might be fixed; then 4 and A will 

 describe similar curves. This property is utilised in Watt's 

 parallel motion (see Art. 271). 



The parallelogram linkage furnishes also a simple instrument 

 for describing ellipses. Let the sides of the parallelogram be 

 23=41= #, 12=34=^; and let a point A ' on 2 3 produced, 

 at the distance b from 2, be fixed (Fig. 63). Then, if i be made 

 to describe a straight line, passing through A f , 4 will describe 

 an ellipse. For, taking A' as origin and A r i as axis of x, we 



Fig. 62. 



A' 



Fig. 63. 



have for the co-ordinates of 4: x=(a-\-2b) cos0, y=asm<j>, 

 whence 



a 2 



267. In the parallelogram 1234, let the link i 2 be turned 

 so as to coincide in direction with 4 3, and then give the links 

 4 i and 3 2 rotations of opposite sense. We thus obtain a link- 



PART I IO 



