27(> Dr. R. S. Ball on the principal Screws of Inertia 



It is to bo observed that the twist involves two elements, 

 i. e. a graphic element (the screw), and a metric element (the 

 amplitude of the twist). A little reflection will show that the 

 constraints which permit displacement from A to B must also 

 admit of any infinitely small twist being made upon the screw 

 defined by A and B. We may therefore say that the body is 

 free to twist about the screw (A, B). If we find on exami- 

 tion that the body cannot be displaced to any position except 

 those which could be attained by twists of suitable amplitude 

 about the screw (A, B), then we can assert that the body has 

 only one degree of freedom ; and that freedom is perfectly ex- 

 pressed by the capacity to twist on one definite screw. 



It is easy to verify in particular cases the general principle 

 that, no matter what the constraints be, a body which has but 

 one degree of freedom can twist about a certain screw and 

 can have no other movements. For example, a body free to 

 rotate about an axis, but not to slide along it, can only twist 

 about a screw of which the pitch is zero, or a body free to 

 slide along an axis, but not to rotate around it, can only twist 

 about a screw whose pitch is infinite. A less obvious instance 

 is presented in the case of a body of which five points are 

 limited each to a given surface ; but even in this case the body 

 is still only free to twist about a certain screw. Draw the five 

 normals to the surfaces, and regard them as the rays of a 

 linear complex. Then the screw about which the body can 

 twist is the principal axis of that complex, while the pitch of 

 the screw is its parameter. These, however, are only illu- 

 strations; our concern is with the general proportion that when 

 a body has but one degree of freedom, from whatever cause 

 arising , it is free to twist about one screw, and only one. 



Suppose, however, it were found that the body, besides 

 being able to twist about a certain screw a, was also able to 

 twist about a second screw (3, then twisting about an in- 

 finite number of other screws must also be possible. For 

 the position attained by a twist about a, followed hy a twist 

 about /3, could have been reached by a single twist about some 

 screw y. Now as the amplitude of the twists about a and /3 

 are arbitrary, it is obvious that y must be one of a singly in- 

 finite number of screws which include a and /3. It follows 

 that the body must be able to twist about all the screws which 

 constitute the generators of a certain ruled surface. 



This surface is called the cylindroid : it is of the third order, 

 its equation being 



z(x 2 + y 2 ) — 2 mxy = 0. 



The cylindroid is already well known to the students of the 

 linear geometry of Plucker. 



