125 



res <>t lu'licoidal motion in such infinite systems is governed by 

 tactly tin >;uni laws, as were previously found in the case of the 

 inbination of ordinary axes of rotation: in general we can deal with 

 surh helicoidal axes in just the same way as if they were mere axes 

 of rotation: the periods 

 of tin- In licoidal axes 

 l>-sible in infinite sys- 

 tems, can also be no 

 other than we found for 

 the axes of simple ro- 

 tation. 



If rotations or helicoi- 

 dal motions be combined 

 dth a translation t per- 

 pendicular to the axis 

 under consideration, it 

 can be easily demonstra- 

 ted (fig. in) that the 

 result of this is always 



a motion about another axis parallel to the first. Let A^ (fig. ///) 

 be an axis of rotation or of helicoidal motion, and let t be the charac- 

 teristic translation perpendicular 

 to that axis. A point P x of the 

 system arrives at P 2 by the 

 rotation through an angle a, 

 round the axis A lt or in P' 2 

 situated above the plane of rota- 

 tion, if A l be a helicoidal axis. 

 Because P x and P 2 are two points 

 of the system nearest to each 

 other, PiP 2 is a characteristic 



Fig. no. 



translation of it, and as / has the 

 same function, P t can always be 

 chosen in such a way that P^j 



H is parallel and equal to t\ this is 



Fig. in. the case represented in fig. 111. 



Now this translation brings P, 



back in P x , and makes A l coincide with a similar axis A{. Therefore 

 the combination of both motions is equivalent to a rotation about 

 an axis passing through P lt which brings, A l into A(; and the angle 





