70 



CuFeS 2 , and one of the numerous forms oicalcite: CaC0 3 . In both 

 cases it may be seen that really the principal axis, although as 

 an axis of the first order only having a period of 180, or 120 

 respectively,, is at the same time an axis of the second order with 

 characteristic angles of 90 and 60. 



And moreover it is also clear from these figures that in the case 

 of calcite there is a real centre of symmetry, which on the contrary 

 is absent in the case of chalcopyrite. The case of Grovea pedalis, as 

 evidently belonging to the group Z)^, we have drawn attention to 

 before. x ) 



11. The last groups which remain to be traced, are those which 

 relate immediately to the three possible endospherical groups T, K, 

 and P, previously dealt with. Again we have to investigate what 

 will be the result of their combination with S H , S v , S D , and /. 



In connection with our reasonings in the case of the analogous 

 deductions from the group D n , and bearing in mind that the groups T 

 and K also possess among their characteristic operations a number 

 of rotations round three binary axes which are perpendicular to 

 each other, we may conclude in the same way as before that only 

 the combinations with S H and S D will produce two different new 

 groups in the case of T. 



For S H and S D combined are equivalent to a rotation through 180 

 round an axis which bisects the angle between two of the above 

 mentioned axes ; this new binary axis is not present in T, but in K its 

 direction is the line joining the middles of two opposite edges of 

 the cube. Therefore the three new groups appear to be: T H , T D , K H ; 

 other ones are not possible. 



With respect to the pentagonal dodecahedral group P, we find 

 in quite the same way, that if the axial system of P should coincide 

 with itself by the added operations of the second order, this addi- 

 tion can be executed only in such a way that the plane of reflection 

 passes through two quinary, two ternary, and two binary axes 

 at the same time. If one of the quinary axes is put in a vertical 

 position, we can regard this added plane as S v ; moreover it will 

 bisect the angle of two pairs of other quinary axes, of two pairs 



!) Of course the groups of the second order, which are related to D n can 

 be deduced as well from the groups C n of the second order, by combining 

 those with binary axes; just in the same way as in the previous chapter we 

 have derived D n from the cyclic groups C n - This however may be left to 

 the reader. 



