824 REPORT— 1901. 



In the given group of operations draw a Jiomogeneous plane in the 

 manner defined above ; ^ this plane will, since the translations of the 

 group are not infinitesiraal,^ and develop space-networks, obey the law of 

 rational indices. 



Apply to the plane thus drawn the operations of the group ; the 

 result is the generation of a system of planes symmetrically distributed 

 through space, all of which are similarly related to the structure regarded 

 as without limits. If mirror-image repetition be not found among the 

 operations of the group, this similarity will amount to identity ; if, on the 

 other hand, enantiomorphous operations are present, the planes will form 

 two equally numerous sets, the relation of the one set to the whole being 

 enantiomorphously similar to that of the other set."' 



Since all the components of the operations of the group which are 

 mere translations are without effect on orientation, the number of 

 different orientations presented by the planes will be strictly determined 

 by the remaining components, and therefore limited.'* As the component 

 operations of the given group which affect orientation are those charac- 

 teristic of some one of the thirty-two classes of crystal symmetry,' the 

 number of orientations presented in the given case will be the same as in 

 such class ; •* i.e., there will be as many infinite sets of parallel planes as 

 there are different orientations. The planes of each set, since they have 

 to obey the translations found in the group, will be equidistant. Among 

 the 230 different types, there are many in which it is possible to select 

 from the set of planes one of each orientation in such a way that the 

 planes selected enclose a space, but in some only of the types thus charac- 

 terised can the planes be so chosen as to outline a symmetrical polyhedron 

 whose axes are axes of the system ; for the remainder centred enclosures 

 of this symmetrical character are impossible.^ 



With the aid of the above conception of a system of similar planes it 

 is not difficult to verify the following propositions : — 



1. The application of an additional movement or enantiomorphous 

 operation to a group, provided the system of axes, planes of symmetry, 

 and other features essential to the group are brought to coincidence 

 {Decku7ig) by this new operation, will lead, when the latter is completely 

 combined in every possible way with those previously present, to the 

 evolution of a derived Schonflies group of operations.* This derived 



' See above, p. 312. The direction of the plane is not. to be a specialised one, 

 except so far as premised by tbe definition : this will ensure that every operation of 

 the group shall effect a change of position of the plane. 



- See Krystallsystcmc unci KrTjstallstructur, pp. 3G0 and G3G, and Proposition (1) 

 above. 



^ Cf. Krydallsyitteme mid Krystalhtruetur, pp. 361, 362. 



■• Ih., p.'3G3. Cf. Prop. (6) above. 



= Cf . Ih., pp. 3G3-3G4, 599, and 637. 



" Cf. Prop. (7) above and Pliil. Mag.^ 1901, series G, i. p. 21. As is the case in 

 some of the latter, planes inclined at 180° will be distinguished from one another, 

 the two sides of a plane being discriminated 



' Cf. p. 31. As all the existing evidence as to the ultimate relative situa- 

 tion of crystal faces concerns tlwir direction only, the question whether iu a given 

 system of similar planes regular polyhedral cells are present or not does not; 

 as yet affect the crystallographer. 



" Krystallsysteme vnd Krystallstructur, p. 383. Schonflies sums up his method 

 in the following fundamental proposition : ' Liisst sich die Punktgriippe G durch 

 Multiplication einer Gruppe G, mit einer Operation 2' erzeugen welchc das Axen- 



