398 
a skeleton cube whose face — fig. 27, in the position 
shown in (fig. 11, Plate II.). 
63. Since (fig. 31) G 2 d x cuts AM in M, and d 5 o 1 cuts AC 1 
also in M, and comparing this with fig. 11, Plate II., we see 
that every face of the twenty-four-faced trapezohedron cuts 
one cubical axis at a distance equal AG V and two other cubical 
axes at m times this distance. 
Taking AG 1 as unity, we see that the three indices of the 
twenty-four-faced trapezohedron are 1, m, and m. Its symbol, 
therefore, is 1, m, m. 
.Representing m as a fraction by Naumamfls symbol is 
m 0 m ; MillePs h, h, It ; Brooke, Levy, and Des Cloizeau^s oA- 
64. m— f occurs in crystals of galena and garnet; m=f- in 
argentite, gold, and tennantite ; m= 2 in amalgam, argentite, 
analcime, boracite, cuprite, dufrenoysite, eulytine, fahlerz, 
franklinite, fluor, gold, galena, garnet, leucite, pyrite, pyro- 
chlore, sal-ammoniac, sodalite, smaltine, and tennantite ; 
m==f- in perowskite ; m = - 1 in fluor; m — 3 in blende, copper, 
fahlerz, fluor, gold, galena, magnetite, pyrite, perowskite, pyro- 
. chlore, and spinelle ; m=4 in sal-ammoniac and kerate ; m= 5 in 
galena; m = 6 in magnetite; m=10 in magnetite; m = 12 in 
blende; m= 16 in galena and magnetite; m=40 in pharma- 
cosiderite. 
65. To find the ratios of the rhombohedral and octahedral axes 
of the twenty -four-faced trapezohedron to those of the 
circumscribing cube . 
The right-hand side of the (fig. 31, Plate IY.) being the 
same by construction as that of (fig. 37, Plate IY.) for the 
four-faced cube. 
Ad x = — AD V or as in § 60. 
1 1+ 2 AD i m+1 
m 
But fig. 31, Ad B =A(L = ’ iU - \/ 2. 
m+ 1 
. a i nr AM m+l m + 1 
tan AdM~ = m — 
Ad 5 m^/ 2 2 
but sin 0 1 Ad 5 =°l&=2- and cos 0 1 Ad,=i^S='^l 
AO l \/ 3 40 ; </ 3 
also sin Ao.d, — sin {180°— (AdJM+O, AdM. 
= sin (AdJM+Otdds). 
= sin AdfiM cos O^c^ + cos Ad 5 M sin O x Ad 5 . 
