124 



represent therefore hdicoidal motions, -- are indeed operations of 

 essential significance for unlimited systems, as we have seen in 

 Chapter //. 



Owing to the fact that in these unlimited systems there are sets 

 of parallel axes of rotation or helicoidal motion, it is of interest to 

 point here again to the fact that the simultaneous existence of such 

 parallel axes involves always the existence of others, which can be 

 found by the construction of Euler (see Chapter ///, p. 28). Some 

 examples may make this clear. 



Let (fig. 109) A T and A z be two parallel quaternary axes. If 

 we apply Euler's construction to find the resulting axis, we must 

 realise that the centre of the 'sphere used in fig. 18 is now at 

 infinite distance, the surface of the sphere therefore being changed 

 into a plane perpendicular to A t and A z , and thus coinciding with the 



plane of our drawing here. 

 When the rotations are both 

 clockwise, we must construct the 



a, B 



angles (= 45) as indi- 

 cated in the figure, and because 

 <C A A 3 A 2 = 90 therefore, it 

 appears that A 3 is a binary axis 

 (7 180), parallel to ^4 x and 

 A z . Indeed, the existence of 



such parallel binary axes, as a necessary consequence of the presence 

 of A! and A z , is confirmed for instance in the patterns of fig. 106, 

 ioy, Ji2, etc. ; the arrangement of the quaternary axes of the pattern 

 appears the same as that of the alternating binary axes. In the same 

 way it is seen from fig. nj, that the senary axes alternate with sets 

 of ternary and of binary axes there, which follow from the simulta- 

 neous presence of the parallel senary axes in exactly the same way. 

 If, however, the rotations round A t and A 2 had opposite directions, 

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

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

 therefore be a translation. From fig. no, which show the successive 

 rotations round A 1 and A z over angles ot, and a, which are together 

 equivalent to a translation A^A'^, it is easily seen that the dimension 



of this translation is 2 A^A^ sin ( ). 



\4 ' 



A detailed study teaches moreover that the combination of 



