538 
NOTES ON POLYHEDRA. 
[375 
the group consists of the 4q symbols 
±1, + ©,...+ 03- 1 ; ± 0> a , ± <f> 2 , ... ± 
(to verify that this is so, it is only necessary to form the equations 
©»©«=©»+«, 3>/=-l, ©^g = <J) s+r , <J)S@r _ <pg_ rj <P r <J> s = _ ©'-^ 
which are at once seen to be true). 
15. The ± general case A gives merely the group of the 2q symbols 
±1, + ©, ... ± ©5“ 1 , 
which has been already mentioned. 
16. The tetrahedron gives the group of 24 symbols, 
\ (± 1 ± i +j ± k) 16 cube roots of + 1 
± h ± j, ± k 6 square „ „ „ 
±1 2 terms 
24 
(the signs ± being all independent). 
17. The cube and octahedron give the group of 48 symbols 
^l (±1±i) ’ ti (±1±j) ’ 
¿2 (±1±i) 
12 fourth roots of + 1 
i(± 1 ± ±j ± k) 
16 cube „ „ „ 
±i ±j, ±k, ^=(±j±k), 
h (±k±i) ' k (±i±j) 
18 square „ „ „ 
±1 
2 terms 
48 
(the signs + being all independent). 
18. The dodecahedron and icosahedron give the group of 120 symbols 
± 
+ 
± 
Vs ± l 
4 
Vs + 1 
4 
Vs ± l 
4 
±\j± 
±P± 
± 2 ^ ± 
V6 + 1 
4 *’ 
Vs +1 . 
Vs +1 . 
4 
h 
\ 
48 fifth roots of + 1 
i (± 1 ± i ±j ± k)