ON THE STRUCTURE OF CRYSTALS. 313 



repetition of posnible parts, each type having its own characteristic group 

 of coincidence-movements. Jordan, however, left his work incomplete 

 and omitted many of the types, which were subsequently discovered by 

 ISohncke, to whom reference must next be made. 



The important bearing of Jordan's work on crystal structure seems to 

 have been entirely overlooked until the publication of the widely influential 

 Avorks of Leonhard Sohncke.^ This writer, employing Wiener's principle - 

 and using Jordan's method to discover what variety of types of symmetry 

 can exist in systems produced by the identical repetition of finite parts or 

 atoms ' throughout space, obtains what he calls a ' regular point-system,' 

 which he thus defines : ' A regular pcint-system is one in which the 

 pencils of lines drawn from each point of the system to all the remainder 

 are congruent with each other.' ^ These systems, if classified according 

 to the position and nature of their axes of symmetry (whether screw-axes 

 or axes of rotation), are sixty-five ' in number. They may conveniently be 

 designated ' Sohncke systems.' 



A Sohncke system then consists of a homogeneous assemblage of 

 points symmetrically and identically arranged about axes of symmetry, 

 and these may be screw- axes such that the points surround them in a 

 spiral arrangement. It might at first sight appear that the latter are 

 inconsistent with the law of rational indices. Since, however, among 

 the coincidence-movements of the system the translations and rotations 

 proper to some space-lattice are always present, it may be proved that 

 a Sohncke- system consists in general of two or more congruent space- 

 lattices which interpenetrate. The translation movements of the Sohncke- 

 system are those which are common to the constituent space -lattices. 



' ' Gruppirung der Molekiile in den KrjstalleTi : eine tlieoretische Ableitung der 

 Krystallsysteme,' i-'w/;/. J.w%., lS()7, csxxii. 75; ' Die unbegr. regelm. Punk*"-s_ysteme 

 als Grundl. e. Theorie der Krystallstructur,' Verh. naturw. Vvr. KarlsniJw, 1870 (7); 

 ' Zuriicliweis. e. Einwurfs geg. d. neue Theor. d. Krystallstruct.,' ^\k'd. Ann., 1879, 

 vi. 545 ; ' Ableitnng d. Grundges. d. Kryst.allsys. a. d. Theor. d. Krystallstructur,' 

 ih., 1882, xvi. 489, and Verh. oiaturw. Vcr. KarUrulie, 1882 (9); ' Elementares 

 Nachweis einer Eigensch. parallelep. I'unktsysteme,' Zciis. Kryst. Min., 1888, xiii. 

 209 ; ' Entwickelung einer Theorie der Krystallstruktur,' Leipzig, 1879 ; ' Ube- 

 Spaltungsflilchen und nat.iirliche Krystallflilchen,' Zeits. Kryst. Min., 1888, xiii. 

 214-235; ' Erweiterung der Theorie "der Krystalle,' ih., 1888, xiv. 426-446; ' Di- 

 Entdeckung des Einlheilungsprincips der Krystalle durch J. E, C. Hessel,' ih., 1890, 

 xviii. 486-498; 'Die Structur der optisch drehenden Krystalle,' ib., 1891, xix. 

 529-559 ; ' Die Structur der hemimorph-hemiedrischen, bezw. tetartoedrischen 

 drehenden Krystalle,' ih., 1896, xxv. 529-530. 



- Solmcke, speaking of his own principal treatise, says : ' Man findet liier die 

 ganze Mannigfaltigkeit der iiberhaupt moglichen Krystalistrukturformen aus einem 

 einzigen Princip, niimlich aus dem selbstverstilndlichen Grundsatze von der regel- 

 miissigen Anordnung, auf streng mathematischem Wege abgeleitet ' (Untwickelunff 

 einer Thecrie der Krystalhtrulifvr, Vorwort, p. iii). 



' //-I. p. iii ; also p. 26: 'Mit. Benutzung des Grundgedankens der Jordan'schen 

 Methode, aber mit Weglasssunsr alles dessen, was nicht direkten Bezug zur Krystall- 

 struktur hat, sind nun im Folgenden alle iiberhaupt moglichen regelmiissigen 

 Punktsysteme von unbegrenzter Ausdehnung abgeleitet und somit alle denkbaren 

 Strukturformen krystallisirter Korper ermittelt.' And later, p. 29 : ' . . . die 

 verschiedenen Arten von Deckbewegungen als Eintheilungsgrund fiir die regel- 

 miissigen Punktsysteme dienen.' He employs some well known kinematic proposi- 

 tions relating to rigid systems to aid him in arriving at his results. 



* Ih., p. 28. 



^ In his principal work, Bie Entmiclielung, &c., Sohncke describes sixty-six 

 types, but subsequently concludes that there are but sixty-five, Nos. 9 and 13 of hig 

 systems being the same type. Zclts, Kryst. Min., 1888, vol. xiy. p. 433. 



