438 
1 
COS (i)~ 
_2_ 
mn 
( See § 107.) 
1+ -- 3 +^ 
m w 
Or by § 110, 
cos a> = - cos 2h cos Ps “ cos Pi cos Pz + cos Pi cos Pi 
= cos 2 2h — 2 cos P 2 cos Pz- 
Which may be computed at once by Byrne's dual logarithms, 
or thus adapted for ordinary logarithmic computation. 
cos co = cos 2 2h 
{- 
2 COSJ92_COSJ9 3 
COS 2 p x 
Let tan a 
2cos_p 2 cos ft* _ cos p 2 cos_p 3 
COS 4 f x 
cos 60 cos 3 
x cos 3 cos (a + 45)° 
Then cos w = cos 3 p 1 (1-tan a)- cos a s i n 45° " 
138. Limits of ilie Form of the Six-faced Tetrahedron. 
As m and n approach in magnitude to unity, the six-faced 
tetrahedron approximates to the tetrahedron. When m — n — 1 , 
the six-faced tetrahedron becomes the tetrahedron, the points 
WWW* and W» (fig. 21, Plate III.) coincide with the 
points * VX 0 6 , and 0° (fig/l5). 0 x TF 4 and CW 2 become 
the straight line 0 2 0 4 , &c., and the six faces round each point 
Op o 3 , Oqj and o 8 lie in the same plane. 
As m and n increase in magnitude greater than unity, and 
also in equality to each other, the six- faced octahedron approxi- 
mates to the cube. When m and n are both infinitely great, 
it coincides with it. In this case each of the four faces which 
meet in the six points G v C 2 , G v &c., G Q , lie in the same plane. 
As m approaches to unity, while n increases in magnitude, the 
six-faced tetrahedron approximates to the rhombic dodeca- 
hedron. When m— 1 and n=cc it becomes the rhombic 
dodecahedron, and the two faces which lie on each side of the 
twelve lines W 2 o v W^o v W 5 o v &c., lie in the same plane, and 
the Go and GW become equal. _ . 
When m equals unity, while n remains finite, the six-laced 
tetrahedron becomes the twelve-faced trapezohedron, and the 
faces od each side of the twelve edges W 2 0 Y he m the same 
plane, but the edges Go and GW are not equal. 
When m and n are equal to each other, both finite and greater 
than unity, the six-faced tetrahedron becomes the three-faced 
tetrahedron, and the faces on each side the twelve lines 
G o,, G 2 o v &c., lie in the same plane. W coincides with 0 and 
WGW becomes a straight line. When m remains finite, and 
n becomes infinite, the six-faced octahedron becomes the four- 
faced cube, and its scalene triangles become isosceles. 
From the above it follows that the cube, rhombic dodeca- 
