51 



1 1 m. iv lx?come clear from these instances also, that the total number 

 non-equivalent operations of the second order which are present 

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

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

 selves, in the sense in which the word "group" was <1< -tin. (! in 

 jnvvious chapter. The evidence of this will be seen from the 

 pie observation that every two operations of the second order 

 together equivalent to some rotation which belongs to the 

 icteristic ones of the group. The number of the operations of the 

 :ond order which are non-equivalent, can therefore be neither 

 iter nor smaller than the number of non-equivalent rotations, 

 thus must be equal to it. That these rotations themselves more- 

 rer form a closed group if the system be a finite one, is so obvious 

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

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

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

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

 :ond order. 



3. Now we can draw from all this a very important conclusion. 

 Q be an arbitrary operation of the second order, characteristic 

 >r the group considered; A(x),. . . .A(p), etc., may be its non-equi- 

 ilent rotations. If we combine all those rotations successively 

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

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

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

 le equal number of rotations, constitute the complete group of 

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

 the second order characteristic of the group, the result would 

 ive been precisely the same; the only difference would appear 

 the succession of the non-equivalent operations of the second 

 rder, as it would result from the second mode of combination. 

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

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

 "m-equivalent rotations successively with one and the same 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 in the preceding chapter. 

 It was this theorem we already mentioned in the beginning of 

 this chapter, and which in future we shall also make frequent use of. 



