437 
0# 
k 5 3 1 Miller; |-( b 1 b^ b^) 
|-(1 5) Naumann; 
Brooke ; in crystals of boracite. 
By the construction fig. 35, the ratio may be readily 
AO, 
determined by plain trigonometry, just as the ratio 
■ AO, 
in § 73. 
It can also be readily determined by geometry of three 
dimensions. For (fig. 22, Plate III.) W 2 is a point in each of 
the three planes Gyyyd^ G 2 o x d 2i C^o s d 3 . 
Isow the equation to the plane 0 1 o 1 c? 2 referred to rectangular 
co-ordinates, AG V AG 2i AG S , is 
1 
was 
-+A + 
m n 1 
To the plane C f 3 o 1 c? 2 is -+JL + . 
1 n 1 
To the plane 0 1 o 3 cZ 3 is— - 
n m 
(A) 
(B) 
(C) . ( See fig. 31*, 
V +±= i 
n m 
x £+*= l 
and fig. 32*, Plate IV.*) 
And since x, y, z will be the same for the point TF 2 where these 
planes meet, 
+i\ = 0, 
n m 
(A) -(C) * P+bW- 
\m n j \n 
Therefore x— —y. 
Also (A-B) 2 (i-l) + * (l_I\=o 
And x=z. 
1 
j+I-I 
m n 
B u t A W 2 = x 2 -\-y 2 + z 2 = 
And A I!', | gAp = 
1 +-- — 
m n m n 
Again, let w be the angle which the normals of the faces 
GiO x d 2 , G^o s d 3 make with each other, or 180° — w be the angle 
ot inclination of the two faces of the six-faced tetrahedron (fig. 
21, Plate III.), oyer the edge G 1 W 2 . 
Then since mn 1 is the symbol of 
and —n—m 1 that of G^d^ 
