GENERAL PRINCIPLES 



15 



an instantaneous centre. The intersection of the perpendiculars 

 to the paths traversed furnish that centre. Let the straight line, 

 AB, move to A'B' (fig. 26). The paths are AA' and BB', 

 and the perpendiculars drawn from their centres intersect at the 

 centre of rotation R. If the final position (2) is such that A'B' 

 is parallel to AB, the perpendiculars of the trajectories will also 

 be parallel ; and the centre of rotation will be at infinity ; so that 

 a rotation in these conditions is equivalent a translation from 

 AB to A'B' (fig. 27). 



Fie. 26. 



R 



Fin. 27. 



The example of the cycloid, in which continuous displacement 

 brings into existence successive centres of rotation, justifies the 

 name of instantaneous centre. 



Finally, if one considers a solid as a pile of plane figures, it will 

 be understood that by a similar rotation, the whole solid can pass 

 from one position to another, all the centres of rotation being 

 on the same line called the instantaneous axis of rotation. As is 

 evident, that instantaneous axis may be at infinity, in which 

 case there is a translation of the solid. In general it may be said 

 that any movement of a solid in space can always be effected 

 by translation, followed by a rotation ; as in the case of a 

 helicoidal movement ( 7) such as a screw being turned in its 

 nut. 



9. Jointed Systems. To transform a movement of one nature 

 into another, we have recourse to jointed systems. It is sufficient 

 to mention the example of the crank which transforms a 

 reciprocating rectilineal move- 

 ment into a continuous circular 

 movement, and vice versa. The 

 rotation is produced round the 

 axis O (fig. 28) by the crank OM 

 jointed to the connecting rod 

 MC, which gives a reciprocating 

 movement to the body CD. It 

 is easily seen that the normals at 

 M and C give the position R of 

 the instantaneous centre of the f<. 28, 



