52 



It may also become clear from these instances, that the total number 

 of non-equivalent operations of the second order which are present 

 in such a group is always the same as the number of rotations which 

 it contains, the last ones always forming a closed group of rotations 

 themselves, in the sense in which the word "group" was defined in 

 the previous chapter. This follows from the simple observation, 

 that every two operations of the second order are together equi- 

 valent to some rotation which belongs to the characteristic ones 

 of the group. The number of the operations of the second order 

 which are non-equivalent, can, therefore, be neither greater nor 

 smaller than the number of non-equivalent rotations, and thus 

 must be equal to it. That these rotations themselves, moreover, form 

 a closed group if the system be a finite one, is so obvious after all 

 that has been said, that it needs no further comment. 



It will also be evident that the whole system of axes and symmetry- 

 planes of the group will be brought to coincidence with itself by the 

 action of every operation of the group, whether of the first or of the 

 second order. 



3. We can draw from all this a very important conclusion. 

 Let Q be an arbitrary operation of the second order, characteristic 

 for the group considered; ^4(#), ^A(pot), etc., may be its non-equi- 

 valent rotations. If we combine all those rotations successively 

 with Q, we shall obtain an equal number of non-equivalent operations 

 of the second order, and as they will bring the whole system of sym- 

 metry-elements to self -coincidence, they will really, together with 

 the equal number of rotations, constitute the complete group of 

 the second order. If instead of Q we had chosen another operation 

 of the second order characteristic of the group, the result would 

 have been precisely the same ; the only difference would lie in the 

 succession of the non-equivalent operations of the second order, 

 as it would result from the second mode of combination. 



It follows from this : that we can derive every group of the second 

 order from one of the first order, by simply combining each of its 

 typical non-equivalent rotations successively with one and the same 

 characteristic operation of the second order Q. 



By this theorem the way is indicated, by which we may come to 

 the complete deduction of all possible symmetry-groups of the second 

 order, starting from those which we met with in the preceding chapter. 

 We already mentioned this theorem in the beginning of this chapter, 

 and we shall also make frequent use of it later on. 



