399 
But in triangle Ao x d 5 = S — 
Ad 5 sin Ao x d 5 
Therefore J.o 1 =? S ; 11 
sin Ao x d z 
sin Ad 5 M 2 
(m + 1) (sin Ad.M cos O x Ad 5 + cos Ad~M sin O x Ad^ 
my/ 2 
(m+ 1) (cos O x Ad 5 + cot Ad 5 M sin O x Ad^ 
m \/2 V 3 
(m + l)(\/2 + J\/2 1 1 
U -+1V3) 
J.0 r 
772 + 1 + l 
1+-+ i 
m m 
Hence the ratio of the rhombic axes of the twenty-four- 
faced trapezohedron to those of circumscribing cube, or 
Ad x 1 
= t y and the ratio of the octahedral axes of the 
i+± 
m 
Ao x 
twenty-four faced trapezohedron, or -^q — 
-+i 
m m 
66. Representing as R v and as i? 2 . 
A.U X AU X 
R x = unity divided by the sum of the reciprocals of the 
first two indices taken in order of magnitude, and 
R . 2 = unity divided by the sum of the reciprocals of the three 
indices. 
When m=-J R x =f and -^ 2 = i 
m=f R L = f R 2 =f 
m=2 E x =f R 2 =\ 
m — ¥ B l-T3 -®2 — IT 
m=f R x =^ R 2 =± 
m= 3 R x = f -R 2 =f 
772=4 R x =± R 2 = f 
m = 5 Ri=% R? — f 
m=10JB 1 =+^ R 2 =f 
772=12^ = 11 R 2 = f 
772=16^=# i? 2 = f 
772=40 R x =^- E 2 =-Jf- 
67. When 772=1, R x =^, and R 2 = 1 h and the twenty-four- 
faced trapezohedron becomes the octahedron. 
2 f 2 
