ON THE 3TRUCTURE OF CRYSTALS. 329 



Fedorow first established the principle tliat in a symmetrical figure 

 the symmetry must be one or more of the following sorts : axis of sym- 

 metry, plane of symmetry and a combination of the two, or composite 

 symmetry (i.e., Curie's alternating symmetry) ; in a regular system of 

 figures, on the other hand, supposed infinitely extended, two more general 

 elements of symmetry are also jDOSsible — namely, a screw axis and a 

 glide-plane of symmetry ; repetition about a screw axis consists of a 

 rotation combined with a translation along the axis ; repetition about 

 a glide-plane consists of reflexion combined with a translation parallel to 

 the plane. The elements of symmetry in a finite figure are simply special 

 cases of the latter in which the translations are zero. 



Like Hessel, Fedorow investigated first the symmetry of finite solid 

 figures in general and then, by limiting the problem by a condition 

 equivalent to the law of rational indices, deduced the thirty -two kinds of 

 symmetry possible for crystals. His method consists practically in com- 

 bining any two of the possible elements and ascertaining to what other 

 elements they give rise : e.g., two axes of digonal symmetry inclined at 

 45° give rise to the axes of a trapezohedral tetragonal crystal ; the total 

 group constitutes a ' Symmetrie art ' or ' class.' 



Two classes are different when in one of them an axis (or, in general, 

 a symmetry element) is present which is absent from the other, or occu- 

 pies a position which it does not occupy in the other. Such a class, 

 therefore, corresponds to a ' group of operations * in the language of 

 Schonfiies. 



A special feature of Fedorow's researches is his analytical expression 

 of the symmetry ; this is described in the second of the above-mentioned 

 memoirs. In this method a point is denoted by an indefinite number of 

 coordinates (although three are suflScierit) — namely, the intercepts made 

 upon all the axes, derived by the symmetrical repetition of one coordi- 

 nate axis, by planes drawn perpendicular to them through the given 

 point. 



Thus, if an axis of p-iold symmetry be taken as one coordinate axis y, 

 and a line perpendicular to it as a second coordinate axis ^/f,, then repeti- 

 tion of 2/0 about 2/ gives ^j — 1 other coordinate axes, ?/], y2> '^'^- -^ point 

 ■whose coordinates are 2/=b, 't/f^=bn, y^=ba then gives rise to a sym- 

 metrical set of points y^=h, yfji=b,, y^=zbs^i, whei-e s may have the 

 different values 0, 1, 2 . . . ^; — 1. By means of equations of this nature, 

 containing also appropriate symbols for repetition about the planes of 

 symmetry, the various sorts of symmetry of figures or of regular systems 

 of figures are deduced and expressed. The method by which they are 

 deduced consists practically in seeking all the possible combinations of 

 the elements of symmetry which are not incompatible with each other. 



In the first memoir, which deals only with the symmetry of finite 

 figures, after establishing all the possible varieties of regular polyhedra 

 and classifying them as isogons (which have similar or symmetrical edges), 

 and isohedra (which have similar or symmetrical faces), and having shown 

 that there are eighteen sorts of typical isohedra,^ Fedorow investigates 

 their symmetry according to the principle that each class (Symmetrie art) 

 corresponds to certain typical isohedra, and, conversely, that when all 

 the typical isohedra are known the various classes of symmetry can be 

 deduced from them. Crystal polyhedra are treated as special cases. 



' A typical isohedron is the figure derived from a polyhedron by moving its faces 

 parallel to themselves until they all touch one and the same sphere. 



