of a Free or Constrained Rigid Body, 279 



expressed by a certain screw-system ; and, on the other hand, 

 we have a group of impulsive forces which constitute a 

 wrench on what may be called the impulsive screw. Now, 

 in consequence of this impulse the body will commence to 

 move ; but the only motion it can execute must be to twist 

 about some screw of its screw-system ; there must therefore 

 be in the screw-system a certain instantaneous screiv corre- 

 sponding to each impulsive screw. If the impulsive screw be 

 given, the instantaneous screw is determined ; but the converse 

 is not true. In fact, the instantaneous screw is limited to the 

 screw-system, whereas the impulsive screw may be anywhere 

 in the universe and with any pitch. It is, however, possible to 

 clear away this ambiguity in a very satisfactory manner. It 

 can be shown that, whenever the body is not quite free, there 

 are a group of impulsive screws corresponding to each instan- 

 taneous screw. It is not here necessary for us to enter into 

 the properties of these groups of impulsive screws, further 

 than to remark that, if a body receive an impulsive wrench 

 upon any one of the screws belonging to this group, it will 

 commence to twist about the instantaneous screw to which the 

 group is correlated. 'In each group of impulsive screws there 

 is one screw which merits most particular attention. It can 

 be proved that in each of these groups there is always one 

 screw (but only one) which not only belongs to the group but 

 also belongs to the screw-system expressing the freedom of the 

 body. It therefore follows that the effect of an impulsive wrench 

 anywhere in space could have been produced by an impulsive 

 wrench on a screw suitably chosen from among the members 

 of the screw-system itself. We need therefore only consider 

 the effect of impulsive wrenches upon screws which actually 

 belong to the screw-system. 



In this way the ambiguity has been dissipated without any 

 sacrifice of generality : to each screw of the system regarded 

 as an instantaneous screw corresponds another screw of the 

 complex regarded as an impulsive screw, and vice versa. 



The study of this system of correspondence between the im- 

 pulsive screw and the instantaneous screw of the same screw- 

 complex, leads to many results of considerable interest, not 

 only on account of their great generality, but also on account 

 of their geometrical character. It is natural to consider, in 

 the first place, whether there are any common elements in the 

 two corresponding systems ; and we are thus led to the dis- 

 covery that ichen a rigid body has freedom of the nth order, then 

 n screws can be selected from the screw-system expressing that 

 freedom such that, if the body receive an impulsive wrench on any 

 one of these screivs, the body will commence to twist about the 

 same screw. These are called the principal screws of inertia. 



