31 



itself at the end. The reduction to the two cases described in the 

 above, takes place by turning every two new planes simultaneously, 

 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. 20). 



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. Hence a figure arbitrarily 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 of this property 

 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 con- 

 sidered as being "kaleidoscope"-figures. However from the teacher's 

 point of view, the method proposed by Viola and Wullf, elegant 



!) 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 

 from each other, but only the "final effect" of their cooperation is considered 



