129 



and P 2 are two points of the system nearest to each other, PiP 2 is 

 a characteristic translation of it, and as t has the same function, 

 P! can always be chosen in such a way that P t P 2 is parallel and 

 equal to t\ this is the case represented in fig. 113. Now this trans- 

 lation brings P 2 back in P lf 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 lf which brings A l 

 into A'; and the angle of rotation of the axis passing through P l 

 must therefore also be a. 



The axis resulting from the simultaneous existence of the trans- 

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



Fig. 114. 



with respect to the rotation-plane of A lt and at the apex 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 l occurs. 



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

 impression of the way in which different motions in such infinite 

 systems, if combined with each other, will determine others. 



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



9 



