402 
Therefore 
Hence the ratio of each rhombic axis of the six-faced octahedron 
to that of the circumscribing cube is -i-p or of unity divided 
' m 
by the sum of the reciprocals of the two smaller indices of 
the six-faced octahedron. 
73. Again in (fig. 35, Plate IV.), in the parallelogram 
CA-DA, 0 1 A= 0 1 H 6 =l,and 0 1 0 1 =AH 5 = v / 2 ; also Ad^Ac^ 
... \/_ 2 
1 +v 
Let y=0 1 AD 5 and (3=Ad 6 M Then Ao^^ 180°- 0 + 7 ). 
But AO f = OjH/ + ADf =1+2=3. 
and A0 1 =\/ §• 
Abo ,rn 
In triangle NAd- cot = 
Also in triangle Ao^J... 
Ao, _ sin (3 _ sin ft 
Jd 6 ~ sin (180— (/3 + y)} sin 0 + 7) 
sin )3 1 
sin J3 cos 7 + cos [3 sin y cos y + cot 0 sin y 
Hence Ho x = 
a/3 
a/2 + 1 
a/ 2 + M 
V3 w \/3 w 
a/3 _ AOi 
(vi)^ + t 
1 +— +- 
n m n 
And 
Ao 1 _ 1 
AO-[ 
l+i+i 
m n 
Hence ratio of the octahedral axis of six-faced octahedron 
is to that of the circumscribing cube as or unity 
1 + -+- 
m n 
divided by the sum of the reciprocals of its three parameters* 
74. Let and B 2 =^ 
