418 
Hence cos 6= cos P 3 cosy> 3 + cos P 2 cos p 2 -bcos P 1 cos p v 
The same formulae which we obtained by geometry of three 
dimensions, § 109. 
121. To find the polar distances and longitudes in terms 
of the indices. 
Referring to § 117 and (fig. 40*, Plate IV.*), G v C 2 , and G s 
are poles of the cube, / is a pole of 1 m n } g of 1 m oo, li of 
n 
1 oo Uj Ic of co 1 — , G 3 of 1 oo oo, G 2 of oo 1 oo, and G x of oo oo 1 . 
J ^3 —Pi* f@ i—JPsy G 2 k=\ v G 3 h=\ 2 , G 3 g=\ 3 . 
Then \ is the distance between the poles of oo 1 11 
and oo 1 oo, that between 1 mn and 1 oo oo. 
Hence, § 107, 
m 
cos 
aA 
COS Py 
1 
1+^3 + ^ 
V <mr n A 
sec 2 X 1 = l + m 
n~ 
tan 3 X j — 
tan X r 
n=m cot X x 
n 2 
m 
n 
•mr 
cy *i I 1 * ^ 
sec- j)j=l + _ + — 
Wj n z 
, 3 1,1 1 A , m 3 \ 
tarn pi = — + — _ll + __ J 
mr n A m* \ n A / 
=-s sec 3 \ 
m z 
tan p x = -1 sec \ 
m 
— cot \ sec \ coi p 1 m = sec \ cot p x 
__ cot ] 0 X 
sin \ 
Again X 2 is the distance between 1 oo n and 1 oo oo, p 2 that 
between 1 mn and oo 1 oo. 
Hence, § 107, 
1 
cos 
Xo“ 
1 
m 
COS 
/x 2 
V'T 
V n z 
1 J 2 i = zr 
A / 1+— 3 + — 
v m A n A 
sec 3 
to 
II 
i+4 
see 2 m 2 ( 1 + -L + A 
n z 
\ m 3 v?J 
tan 3 
K= 
1 
n 3 
= i+ 4 i+ S) 
n = cot X 2 tan 3 p 2 = m 3 sec 3 X 2 
tan p = m sec X 2 
m=tanj? 2 cos'X 2 
Also X 3 is the distance between 1 m oo and 1 oo oo, p 3 that 
between 1 m n and oo oo 1 . 
