126 



of rotation of the axis passing through P 1 must therefore also be a. 



The axis resulting from the simultaneous existence of the trans- 

 lation / and the rotation about A lt is evidently situated normally 

 with respect to the rotation-plane of A lt and in the top of an isosceles 

 triangle which has t as its base, and a as its top-angle ; the top lies 

 at that side of t in the direction of which the rotation round A occurs. 



10. These instances may be sufficient to give at least some 

 impression of the way in which different motions in such infinite 

 systems will determine others, if combined with each other. 



In chapter // we have indicated how the symmetry-properties of 

 such systems can be generally deduced by the method of Boldyrew, 



9- 



o 

 * 9 



t 4 



* t v * . t ' * f 



ft ^ 



4. f 



-3 



Fig. 112. 



and others. The systematical deduction of all possible symmetrical 

 arrangements, being a purely mathematical and very extensive 

 problem, may also therefore be omitted here, and only some. general 

 properties of these systems be elucidated by suitably chosen examples. 

 Most of the sixty-five Sohnckian systems can be imagined to 

 be deduced from the space-lattices of Bravais by replacing each 

 point of them by definite, similarly composed, groups of points, the 

 symmetry-elements of which are however differently oriented with 



