419 
And, § 107, 
cos Ac 
•v/i+i, 
V 
cos y> 3 = 
m A 
n* 
sec 2 A,=l + _ 
m- 
tan 3 X 3 = 
m* 
m — cot A, 
,A=n *( 1+ i + ;?) 
( 1+ i) 
= 1 + n 2 
tan 3 _p 3 = n 2 sec 2 A 3 
^=tan^ 3 cos A 3 . 
Hence the indices being given, the p*olar distances and 
longitudes can be determined, or the polar distances and 
longitudes being given the indices can be determined. 
122. To find the polar distances of any two adjacent poles 
of faces of the six-faced octahedron, or of the supplement of 
the angle over the edge of any two adjacent faces, in terms of 
the indices. 
Let 0 be the angle between any two poles adjacent to the 
arc CO (figs. 31* and 32*, Plate IV.*), <p adjacent to OD, 
and \p adjacent to C D. 
For the faces nm 1, mn 1, 
2 
cos 0 = 
— +— +i 
mn mn 
1 
mn 
/y^('i+I + iVJ_ + 4+i) i+4 j+4 
v V 3 m 2 ) \m 2 n 3 ) ™ ir 
Similarly for 1 nm and 1 mu we have 
— +i 
« mn 
cos 0= y 
1+— g + “2 
m A n* 
The same is true over every arc CO in (figs. 31* and 32* 
Plate IV.*). 
P or the faces m 1 n and 1 mn cos <p = 
1+1+1 
m m n 2 
Jl - 1 
m n 2 
i+l+l 
i+1+l 
m~ n " 
I+i-i 
For the faces m 1 n, m 1 n cos \p= 
m 
n" 
1 + — f- — 
•** 1 o 1 Q 
wr n* 
