30 



until they pass through the point of intersection of the first three 

 planes; etc. 



In considering this, the truth of our previous statement is evident, 

 that the general characteristic symmetrical operation of the first order 

 is the rotation round an ordinary axis or round a screw-axis, and that 

 of the second order is the rotation round a mirror-axis, (p. 19). 



14. From these deductions it will now be clear that all theo- 

 rems concerning motions in space, as described by translations, 

 rotations, and helicoidal motions, may be reduced to a combination 

 of successive reflections in two, three, or four not independently 

 acting, and therefore partially virtual mirror-planes. And by the 

 principle of simultaneously turning every pair of intersecting 

 mirror-planes, with their angle of intersection kept unaltered, through 

 an arbitrary angle round their line of intersection, we can find 

 without much trouble the resulting motion of a stereometrical 

 system, if the composing operations are given. 



Indeed, all theorems of the doctrine of symmetry may therefore be 

 exactly demonstrated in this way, as was indicated by Boldyrew 

 in the paper already referred to. Of this property, that a figure arbi- 

 trarily situated in space can always be made to coincide completely 

 with a figure congruent with the first by a certain combination of 

 successive reflections in no more than three mirror-planes which do 

 not act independently from each-other, nor pass through the same 

 straight line, C. Viola 1 ) and G. Wulff 2 ) have made use to give a 

 systematical deduction of the 32 possible crystal-classes. The rotation 

 round an axis of the first order is in this case always the result of 

 successive reflections in two existent or virtual 3 ) intersecting 

 mirror-planes; the rotation round an axis of the second order is 

 described as the action of three successive reflections in planes 

 passing through one point 0, and of which one is perpendicular 

 to both the others. We can express this result by saying that all 

 finite, symmetrical figures may be considered as being "caleidoscope"- 

 f igures. However from the teacher's point of view, the method proposed 

 by Viola- and Wulff, elegant as it is, has some disadvantages in 

 so far, that in such figures as differ from their mirror-images, such 

 reflections can only have a virtual significance, these figures of 

 course possessing no real planes of symmetry whatever. 3 ) 



1) C. Viola, Neues Jahrbuchf. Miner. Geol., und Pal., Beil. Band 10. 507. (1896). 



2) G. Wulff, Zeits. f. Kryst. u. Miner. 27. 556. (1896). 



3 ) Indeed "virtual" planes of reflection, as they are not acting independently 



