415 
Also eight more faces mnl, Inm, 1 nm, mnl, mn f, 
Inm, Inm , mnl , having the same south polar distance and 
longitudes as the former. 
The 16 other faces will have the similar polar distances and 
longitudes, taking G 3 as the north and C 5 as the south pole, 
and C 1 G 2 G 6 G 4j as the equator. 
117. (Fig. 39*, Plate IV.*). — In the three rectangular cubical 
axes, take AB=1, AM—m, AN=n. 
Through A draw AG perpendicular MB, AH perpendicular 
NG 3 , and AK perpendicular MN. 
J oin NG, HM, and BK meeting in F. J oin AF. 
Since the normal from A or the perpendicular to the plane 
NMB must, by construction, lie in each of the three planes 
NAG, HAM, and KAB, AF, their common intersection, must 
be the normal to the plane NMB. 
Hence AF is the normal to the plane whose notation is 1 m n. 
AG is the normal to a plane passing through MC 3 parallel to 
AY, or the normal to a face of the four-faced cube whose notation 
is 1 m oo, AH the normal to 1 oo n, AK to oomwor oo 1 — • 
m 
(Fig. 40*, Plate IY .*). — Let G v C 2 , G 3 be the poles of the three 
rectangular or cubical axes, or the points where AN, AM, and 
AB of fig. 39* cut the sphere of projection. 
Let h, h, and g be the points where AB, AK, and AG cut 
the sphere of projection. Join G x g, G 3 k, and C 2 hhj arcs of 
great circles meeting in /. 
Then g is the pole of 1 m oo, h of 1 oo n, k of oo 1 — , and f of 
1 mn. 
Let fC 3 = lh , fG 2 =rp 2 , fG x = V3 , G 2 Jc=\, GJi=\, G 3 g=X 3 . 
Then p v jo 2 , and jp 3 will be the polar distances of the pole of 
1 mn from C 3 , C 2 , and G v taken in order of magnitude. 
Comparing § 96 with (fig. 31*, Plate IY .*), the face 1 mn 
cuts the axis AG 3 in B, AG 2 in M, and AG X in N to form 
(fig. 39*). Hence arc Gj (fig. 40*) =p 3 , and G 3 g = \ 3 , is its 
polar distance and longitude. 
The face 1 nm cuts the axis AG 3 in B, AG 2 in N, and AG X 
in M; and (fig. 40*) G 2 f=jp 2 and G 3 h = A 9 , is its polar distance 
and longitude. 
Also the face mnl cuts the axis AG 3 in M, AG 2 in N, and 
AG l in B ; and (fig. 40*) G 3 f=2h an( f G 2 k=\, is its polar 
distance and longitude. 
Calling- (fig g . 31 * and 32 *, Plate IV.*), G x the North pole, 
C 3 G 2 O 5 the equator, and measuring longitude from G 3 , \ 3 will 
be the longitude of 1 m n, 90°~A 3 of m 1 n, 90°H- A. of m 1 v, 
2 g 2 
