DYNAMICS OF A RIGID BODY. 1 1 7 



numerous that a finite number (three) can be drawn 

 through every point in space. In the screw complex of 

 the fourth order a cone of screws can be drawn through 

 every point, while to a screw complex of the fifth order 

 belongs a screw of suitable pitch on every straight line 

 in space. 



1 08. Screw Complex of the Third Order. We shall 

 now consider the collocation of the screws in space 

 which constitute a screw complex of the third order. A 

 free rigid body can receive six independent displace- 

 ments. Its position is, therefore, to be specified by six 

 co-ordinates. If, however, the body be so constrained 

 that its six co-ordinates must always satisfy three equa- 

 tions of condition, there are then only three really inde- 

 pendent co-ordinates, and any position possible for a 

 body so circumstanced may be attained by twists about 

 three fixed screws, provided that twists about these 

 screws are permitted by the constraints. 



Let A be an initial position of a rigid body M. Let 

 M be moved from A to a closely adjacent position, 

 and let x be the screw by twisting about which this 

 movement has been effected ; similarly let y and z be 

 the two screws, twists about which would have brought 

 the body from A to two other adjacent positions. 

 We thus have three screws x, y, z, which completely 

 specify the circumstances of the body so far as its capacity 

 for movement is considered. 



Since M can be twisted about each and all of x, y, s, 

 it must be capable of twisting about a doubly infinite 

 number of other screws. For suppose that by twists of 

 amplitude ^, y, z', the final position V is attained. 

 This position could have been reached by twisting 

 about v y so as to come from A to V by a single 

 twist. As the ratios of x f to y, and 2', are arbitrary, 



