226 The American Naturalist. [April, 
plane, i. e., by points which occupy the angles of a series of 
regular hexagons. Thus, in the adjoining figure the dots form 
a Sohncke system in one plane, since the configuration of the 
system round any one point is similar to that round any other; 
but they do not form a Bravais web, since the points do not lie 
at equal distances along straight lines. 
If, however, points, represented in the figure by the circles 
o, be placed at the centres of the hexagons, they will by them- 
selves constitute a web, and the hexagonal system may be de- 
rived from this web by replacing each of its points by a group 
of two satellites, A and B. Or, from the second point of view^ 
the arrangement may be regarded as a triangular web, con- 
taining the points A, completely interpenetrated by a similar 
web, containing the points B. 
It is a remarkable feature of the Sohncke systems that some 
among them are characterized by a spiral disposition of the 
particles along the threads of a right- or left-handed screw : 
now this spiral character, which does not belong to any of the 
Bravais networks, supplies a geometrical basis for the right- 
or left-handed nature of some merohedral crystals which pos- 
sess the property of right- or left-handed rotary polarization. 
The theory of Sohncke, as sketched above, appeared to be 
expressed in the most general form possible, and to include all 
conceivable varieties of crystalline symmetry. 
It has, however, recently been pointed out by Wulff' that 
the partial symmetry of certain crystals belonging to the 
rhombohedral system — that, namely, of the minerals phenacite 
and dioptase — is not represented among the sixty-five arrange- 
ments of Sohncke. 
Other systems of points in space have also been studied by 
Haag' and Wulff, which do not exactly possess the properties 
of a Sohncke system, and yet might reasonably be adopted as 
the basis of crystalline structure, since they lead to known 
crystalline forms." These, however, and all other systems of 
» Zeiisckr.f. Kryst. xiii. (.S87) p. 503. 
* " Die regularen Krystallkorper." (Rottweil, 1887.) 
3 Cf. W. Barlow, Nature, xxix. (1884) pp. 186, 205. 
