RIGIDITY 91 



64. A rigid body can move about in space without changing 

 the direction of any line in it such a motion is called a motion 

 of translation. It can also turn about any point P without the 

 position of P altering such a motion is called a motion of rota- 

 tion about P. It can again have a motion compounded of a motion 

 of translation and one of rotation, and this we shall show to be 

 the most general motion which it can possibly have. 



65. First we must notice that a rigid body is fixed when any 

 three points in it are fixed, provided that these three points do not 

 lie in a straight line. For let A, B, C be the three points. If we 

 fix A and B, any motion which can take place must, since the 

 body is by hypothesis perfectly rigid, be one in which the dis- 

 tances of any other point P from A and B remain unaltered. 

 Thus P must describe a circle with AB as axis, and the motion 

 of the body must be one of rotation about the line AB. Thus 

 if A, B, C are not in a straight line, C must describe a circle 

 about AB. But if C also is fixed, this cannot happen; in other 

 words, no motion can take place, so that the body is fixed in 

 position. 



Thus the position of a rigid body is determined when the posi- 

 tions of three non-collinear points in it are determined. 



66. We can now prove the theorem : 



TJie most general motion of a rigid ~body is compounded of a 

 motion of translation and one of rotation. 



In fig. 46 let the figure on the left represent the body in its 

 original position, and let the heavy curve on the right represent 

 the body after it has been moved in any way. Let P be the 

 position of any particle of the body in its original position, and 

 let Q be the position of the same particle after the motion has 

 taken place. 



Imagine that the body is first moved from its original position 

 in such a way that the point P. moves to Q, while all lines in the 

 body remain parallel to their original positions. This motion is 

 one of pure translation. After this motion has taken place, we 

 can turn the body about the point Q in such a way as to turn it 



