412 
108. In fig. 83, Plate IV.*, let _p 1 = angle FAB, which the 
normal AF makes with the axis AX; angle FAG, which 
the normal makes with the axis AY; and _p 3 = angle FAD 
makes with AZ. 
AX is the normal to a face of the cube which cuts the axis 
AX at a, AY at oo, and AZ at co ; or a x z=a, ^= 00 =^, and 
C =00=i 
and cos p x = 
a 
V(^+F+?) ii V5+R 
AY is the normal to a face of the cube, or a plane whose 
indices are \=b, and c 1 =- 
COS p 2 : 
/ 1 , 1,1 
V a * + 0 + c « 
AZ is the normal to a plane whose indices are 
and c x = c, 
and cos p 3 = 
V ' 
The same formulae we obtained in § 106. 
1 09. If p lf p 2 , p z be the angles which the normal to the plane 
whose indices are a b c, makes with the three axes AX, AY, 
and AZ ; 
Also, q v q 2 , q 3 the angles the normal to the plane whose 
indices are a 1 \ c v makes with the same axes, ^ 
a b 
Then cos ]pi = \ — ^ ^= =- cos Vi— 
f\ , 1., 1 
V a 3 ' 6 2 + c 2 
and cos p 3 = 
Vi + F - + ? 
a/4+p+4 
