61] 



POINSOT'S CENTRAL AXIS. 



211 



It is evident that if we slide the whole of Fig. 58 along or turn 

 it around the central axis nothing is changed, consequently if we 

 suppose the vector S laid off at 

 every point of space 0, and con- 

 sider the assemblage of couples 

 thus formed, the assemblage re- 

 mains unchanged if we rotate it 

 about or slide it along the cen- 

 tral axis. 



Every S is tangent to a cer- 

 tain helix, or locus of a point 

 which moves on a circular cylinder 



, . J Fig. 58. 



in a path making a constant angle 



with its generators (Fig. 59). This angle is less as the diameter of the 

 cylinders is less, so that 

 E 



10) 



tan# = # , 



All these helices have however 

 one constant in common, 

 namely the distance traversed 

 parallel to the central axis for 

 each turn. If dr be the trans- 

 lation for a rotation do, we 

 have 



xdo 



*s : 



& 



Then 



12) 



I> 



Fig. 59. 



is the traverse for each turn, 



and is called the pitch of the 



helix. Every helix lies on a ruled screw -surface, made by the revolution 



of a line perpendicular to the central axis, which slides along it a 



distance proportional to the angle of rotation, the pitch of the screw 



o 



being p = %TC --JJ- The lines of the assemblage of moments have every 



direction in space there are a triple infinity of lines of the system 

 (one for each point in space), but only a double infiriity of direc- 

 tions therefore every plane cutting all these lines has for rjjfcs 

 points (a double infinity), every possible direction for S. Fpj? Qtye 

 point only is this perpendicular to the plane. This point ,-ii 



14* 



