8 THE THEORY OF SCREWS. [4, 



4. A Geometrical Investigation. 



We can now demonstrate that whenever a body admits of an indefinitely 

 small movement of a continuous nature it must be capable of executing that 

 particular kind of movement denoted by a twist about a screw. 



Let AI be a standard position of the body, and let P be any marked 

 point of the body initially at P lt As the body is displaced continuously to a 

 neighbouring position, P will generally pursue a certain trajectory which, as 

 the motion is small, may be identified with its tangent on which P n is a 

 point adjacent to P,. In travelling from P, to P n , P passes through the 

 several positions, P 2 , . . . P^. In a similar manner every other point, Q 1} of the 

 rigid body will pass through a series of positions, Q 2 , &c., to Q n . We thus 

 have the points of the body initially at P 1} Q lt R 1} respectively, and each 

 moves along a straight line through the successive systems of positions 

 P*, Qa, RZ, &c., on to the final position P n , Q n , R n . We may thus think of 

 the consecutive positions occupied by the body A 2&amp;gt; A 3 , &c., as defined by the 

 groups of points P lt Q 1} R 1 and P 2 , Q 2 , R 2 , &c. We have now to show that if 

 the body be twisted by a continuous screw motion direct from A 1 to A n , it 

 will pass through the series of positions A 2&amp;gt; A 3 , &c. It must be remembered 

 that this is hardly to be regarded as an obvious consequence. From the 

 initial position A 1 to the final position A n , the number of routes are generally 

 infinitely various, but when these situations are contiguous, it is always 

 possible to pass by a twist about a screw from A 1 to A n via the positions 

 A 2 , A 3 . .. A n _!. 



Suppose the body be carried direct by a twist about a screw from the 

 position AT, to the position A n . Since this motion is infinitely small, each 

 point of the body will be carried along a straight line, and as Pj is to be 

 conveyed to P B , this straight line can be no other than the line PiP n . 

 In its progress P l will have reached the position P 2 , and when it is 

 there the points Q l} R t will each have advanced to certain positions along 

 the lines QtQ n and R^n, respectively. But the points reached by Q l and 

 R 1 can be no other than the points Q. 2 and R 2 , respectively. To prove this 

 we shall take the case where P 1} Q 1} R 1 are collinear. Suppose that when 

 P! has advanced to P 2 , Q 1 shall not have reached Q.,, but shall be at 

 the intermediate point Q . (Fig. 1.) Then the line P& will have moved to 

 P 2 Q , and as 7^ can only be conveyed along R^.,, while at the same time 

 it must lie along P 2 Q , it follows that the lines P 2 Q and R^ 2 must intersect 

 at the point R , and consequently all the lines in this figure lie in a plane. 

 Further, P 2 Q 2 and P 2 Q are each equal to PjQi, as the body is rigid, and so 

 also P 2 R and P 2 R 2 are equal to P^. Hence it follows that QiQ and R^o 

 are parallel, and consequently all the points on the line P&Ri are displaced 

 in parallel directions. It would hence follow that the motion of every point 

 in the body was in a parallel direction, and that consequently the entire 



