22 GEOMETRY OF MOTION. [41. 



mon direction of the axes of rotation. To bring any three 

 points of this plane into their final position it is only necessary 

 to give the body a translation at right angles to TT such as to 

 bring TT into its final position and then to add the necessary 

 rotation for plane motion. 



We have therefore the important proposition that it is always 

 possible to bring a rigid body M from any position M into any 

 other position M 1 by a translation combined with a rotation about 

 an axis parallel to the direction of translation, and this can be 

 done in only one way. The axis so determined is called the 

 central axis of the displacement. 



The order of translation and rotation about the central axis is 

 indifferent ; indeed, translation and rotation might take place 

 simultaneously. 



41. A motion of a rigid body consisting of a rotation about 

 an axis combined with a translation parallel to the axis is called 

 a screw motion, or a twist. We have proved therefore, in Art. 

 40, that the most general displacement of a rigid body can be 

 brought about by a single twist. 



42. To construct the central axis and find the translation 

 and angle of the twist when the displacement is given by the 

 positions A^ B Q , 7 and A v B^ 6\ of three points of the body, 

 we first remark that the projection on the central axis of the 

 displacement of any point, say A^A^ is equal to the translation 

 of the twist, and hence the projections of the displacements of 

 all points of the body (such as A^A^ B Q B ly C Q C^) ar.e all equal. 

 If therefore from any point O we draw lines OA, OB, OC equal 

 and parallel to A Q A V B Q B V C Q C V their ends A, B, C will lie in 

 a plane TT perpendicular to the central axis, and the perpendicu- 

 lar p dropped from O on this plane TT will represent in length 

 and direction the translation of the twist. 



The direction of the central axis being thus determined, we 

 find its position in space by projecting the displacements of any 

 two of the three given points, say A Q A l and B^B V on the plane 



