20 GEOMETRY OF MOTION. [38. 



the point has then but one degree of freedom and two con- 

 straints. 



A rigid body that is perfectly free to move has six degrees of 

 freedom. For we have seen that its position is fully determined 

 when three of its points not in the same line are fixed. The 

 nine co-ordinates of these points are, however, not indepen- 

 dent ; they are connected by the three equations expressing 

 that the three distances between the three points are invariable. 

 Thus the number of independent conditions is 9 3 = 6. 



A rigid body with one fixed point has three degrees of freedom 

 and therefore three constraints. For it takes two more points, 

 i.e. six co-ordinates, to fix the position of the body ; and the 

 distances of these two points from each other and from the 

 fixed point being invariable, there are again three conditional 

 equations to which the six co-ordinates are subject. The three 

 co-ordinates of the fixed point may be regarded as the three 

 constraints. 



A rigid body with two fixed points, i.e., with a fixed axis, has 

 one degree of freedom, and five constraints. Indeed, the six 

 co-ordinates of the two fixed points are equivalent to five con- 

 straining conditions, since the distance of these two points is 

 invariable.* 



38. Let us now consider any two positions M Q , M of a rigid 

 body M, given by the positions A Q , B Q , C Q and A v B v \ of 

 three points A, B, C of the body. The displacement M Q M l 

 can be effected in various ways. Thus we might for instance 

 begin by giving the whole body a translation equal to A Q A l 

 which would bring the point A to its final position while all 

 other points of the body would be displaced by distances par- 

 allel and equal to A Q A r As the body has now one of its 

 points, A, in its final position, it will (by Art. 34) require only 



* Interesting remarks on the mechanical means of producing constraints of 

 various degrees will be found in THOMSON and TAIT, Natural philosophy, London, 

 Macmillan, new edition, 1879, Art. 195 sq. (Part I., p. 149). 



