128 



so that the algebraic sum of their angles of rotation were = 0, the 



axis A 3 would be situated at an infinite distance ; the result would, 



therefore, be a translation. From fig. 112, which show the successive 



rotations round A l and 



A 2 over angles & and , 



which are together equi- 



valent to a translation 



A^A\, it is easily seen 



that the dimension of 



this translation is 



2 



sn 



Fig. 112. 



( V 

 W/ 

 A detailed study 



teaches, moreover, that 



the combination of axes 



of helicoidal motion in 



such infinite systems is 



governed by exactly the 



same laws, as were previously found in the case of the combination 



of ordinary axes of rotation: in general we can deal with such heli- 



coidal axes in just the same way 

 as if they were mere axes of 

 rotation: the periods of the heli- 

 coidal axes possible in infinite 

 systems, can also be no other 

 than that, which we found lor 

 the simple axes of rotation. 



If rotations or helicoidal mo- 

 tions be combined with a trans- 

 lation t perpendicular to the axis 



Ai 



under consideration, it can be 

 easily demonstrated (fig. nj) 

 that the result of this is always 

 a motion about another axis 

 Fig. 113. parallel to the first. Let A l (fig. 



nj) be an axis of rotation or 



of helicoidal motion, and let t be the characteristic translation per- 

 pendicular to that axis. A point P 1 of the system arrives at P 2 by 

 the rotation through an angle # round the axis A ly or in P\ situated 

 above the plane of rotation, if A l is a helicoidal axis. Because P x 



