37-] SCREW MOTION. ! 9 



35. If we consider a series of positions of the moving figure, 

 F , F v Fy . . ., we obtain a series of axes of rotation, say c v c z , . . . ; 

 and in the limit when these positions follow one another at 

 infinitely near intervals, the axes c v c y ... will form a cone fixed 

 in space, with the vertex at the centre O of the sphere. The 

 points C v C 2 , ... where these axes intersect the sphere form a 

 curve (c) on the fixed sphere, while the points C' v C'%, ... of the 

 moving figure with which these fixed points come to coincide 

 form a spherical curve (c') invariably connected with the moving 

 figure. The whole motion may be produced by the rolling of 

 the curve (c'} over the curve (c), or also by the rolling of the 

 corresponding cones one over the other. We have thus the 

 proposition that any continuous motion of a rigid body having a 

 fixed point can be produced by the rolling of a cone fixed in the 

 body on a fixed cone, the vertices of both cones being at the fixed 

 point. 



IV. Screw Motion. 



36. The position of a rigid body in space is fully determined 

 by the position of any three of its points not situated in the 

 same straight line. For any fourth point of the body will form 

 an invariable tetrahedron with these three points. As two 

 points determine a straight line, the position of a rigid body 

 may also be given by the position of a point and line or by 

 the positions of two intersecting or parallel lines of the body. 



37. The position of a point being determined by its three 

 co-ordinates requires three conditions to be fixed. A point is 

 therefore said to have three degrees of freedom when its position 

 is not subject to any conditions. One conditional equation 

 between its co-ordinates restricts the point to the surface repre- 

 sented by that equation ; the point is then said to have but 

 two degrees of freedom and one constraint. Two conditions 

 would restrict the point to a line, the curve of intersection of 

 the two surfaces represented by the equations of condition ; 



