278 Dr. R. S. Ball on the principal Screws of Inertia 



describing, then the system of screws is quadruply infinite ; in 

 fact, ihe body can then twist about one screw of a certain pitch 

 OD every line in space. 



Finally, if the body can twist about six screws not belong- 

 ing to the system just mentioned, then the body has freedom 

 of the sixth order, and is, in fact, perfectly free. 



We thus see that, corresponding to each order of freedom, a 

 certain group of screws is appropriate ; and we may call such 

 a group a screw-system for the sake of brevity. Thus, in the 

 case of freedom of the second order the screw-system is a cylin- 

 droid ; in the case of freedom of the fourth order the screw- 

 system consists of all the screws of proper pitch which inter- 

 sect a generator of a cylindroid at right angles, and so on. 



By this preliminary investigation we are enabled to dismiss 

 entirely all further mention of the constraints. Every con- 

 ceivable form of constraints can only give the body permission 

 to twist about one of the six types of screw-system. I have 

 not in this brief summary attempted to give any demonstra- 

 tions of the different theorems involved. For these, reference 

 may be made to the ' Theory of Screws ' *. 



We have now laid the foundation of the first part of the 

 problem to be discussed in this paper, inasmuch as we have 

 shown how the body may move ; the next question is to ascer- 

 tain how the body will move when it receives an impulse. 



It willj however, be first necessary to consider the most 

 general form of impulse which the body can receive. Now 

 it is well known that all the forces acting upon a rigid body 

 may be reduced to a single force, and a couple in a plane per- 

 pendicular to that force. The efficiency of the couple is ex- 

 pressed by its moment ; and the moment is the product of a 

 force and a linear magnitude. It will not be unnatural to as- 

 sociate this force and couple with the conception of the screw, 

 already introduced. We may use the expression wrench to 

 denote a force along a screw and a couple in a plane perpen- 

 dicular to the screw, the moment of the couple being equal to 

 the product of the force and the pitch of the screw. Thus, 

 every system of force acting upon a rigid body constitutes a 

 wrench upon a screiv ; and it is completely determined when the 

 screw on the one hand, and the force on the other, are both given. 



The analogy subsisting between the twist and the wrench, 

 as implied by their mutual connexion with the abstract geo- 

 metrical conception of a screw, will be the main source of the 

 theorems now to be enunciated. 



The original problem has now been brought into this con- 

 dition. On the one hand, we have a body whose freedom is 



* Ball's ' Theory of Screws :' Dublin, 1876. 



