73 



u pollen-, rll- ill' some Polygoneae, according to II;n-. kel'a d 

 1 1< '\\i-\vr it IN tlifticnlt to say whether such individuals belong 

 ly t<> tin- class, or only have the symmetry of the groups 

 ami I\ themselves, ll so, tin- drawings of fig. 49 may be included 

 V the instances just mentioned may be among those given 

 tin- preceding chapter. 



13. No other symmetry-groups than those deduced in the preceding 



re possible for finite stereometrical figures, as long as axes of isotropy 



re not concerned. The whole investigation therefore has led to the 



suit that the different types of symmetrical figures are only few 



number, although of course their total number is infinitely great, 



jcause n can have all possible values. 



If we review these principal types here once more, we shall find 

 le following result: 



A. Symmetrical figures which differ from their mirror-images. 



1. Cyclic groups C n 



2. Dihedron-groups: D n 



j. Endospherical groups'. T, K and P. 

 All figures belonging to A may exist in two enantiomorphous forms. 



B. Symmetrical figures, which are identical with their mirror-images. 



4 . Cyclic groups of the second order : C n ', special cases : S and 7. 

 jv The groups: C v n and C%. 



6. The groups: D% and D%. 



7. The groups: T H , T D , K u , and P 7 . 



The number of these different types does not exceed fourteen or 

 ixteen ; for finite figures this exhausts the possible symmetries if n 

 all values from / to infinitely great. The groups with axes of 

 )tropy (n oo ) will be dealt with in detail in the following chapter. 



>ome Literature on the General Theory of Symmetrical Arrangement: 



1. J. F. C. Hessel, Krystallometrie, oder Krystallonomie und 

 Krystallographie, auf eigenthiimlirhe Weise und mit Zugrunde- 

 legung neuer allgemeiner Lehren der reinen Gestalltenkunde 

 bearbeitet: Gehler's physik. Worterbuch 5, 1023. (1830); Ost- 

 wald's Klass. der exakten Wiss., Ed. Hess, 88 and 89. (1897). 



2. A. Bravais. Memoire sur les polyedres de forme symetrique; 

 Journ. de mathem. p. Liouville 14. 141 (1849) Ostw. Klass. 

 d. e. Wiss. 17. (1890); Ed. Groth; Etudes crystallographiques, 

 (1886). Gauth.-Villars, p. XXXI 1. 



3. A. Gadolin, Deduction of all crystallographical Systems and 

 their Subdivisions by means of a single General Principle; Yerh. 



