41 G 
v 3 
min 
]80°— X 3 of Tto n, 180°+X 3 of Imn, 270°-X 3 of min, 
270°+X, of ml n, and 360°— X 3 of Itom. 
The north polar distances of these eight faces will ^lb® Pr 
X 3 the longitude of 1 to n, 90°-A 3 of m 1 n, 90 +X 3 _of 
180°— X 3 of I mn, 180°+X 3 of Imn, 270°-X 3 of m 1 n, 
270° + X, of ml n, and 360°— X 3 of 1 mn. 
The north polar distances of these eight faces will each ho 
18 L will' be the longitude_of lnm, 90°-X 2 o <t_n 1 m, 90° + X 2 
of nlm, 180°— X 2 of lnm, 180° + X 2 of lnm, 270 -X 2 
of nl m, 270° + X 2 of n 1 m, 360°-X 2 of lnm. 
The north polar distances of these eight faces will each bep 2 . 
The eight similar faces in the southern hemisphere will 
have the same longitudes as those corresponding to them m 
the northern, the eight north polar distances being each equal 
^gQO ^ 
X, will’ be the longitude of mnl, 90 °-/i_° f ” ™ 1 ’ o 90 ^ + ^ 
of it to 1, 180°-X 1 of mnl, lSO 0 -^ of mnl, 270 -Xj ot 
nml, 270° + X 1 of nml, and S 60 o -\ ot mnl. . 
^ will be the north polar distance of each of these eight 
faces 
The corresponding eight faces of the southern hemisphere 
will have the same longitudes as the corresponding ones m 
the northern, 180°-p, being the north polar distance of these 
elg Hence e the 48 faces or poles of the six-faced octahedron can 
be expressed in terms of p„ p v X 2 , and p 3 , X 3 ; and, as a 
other forms of the cubical system can be derived from those 
of the six-faced octahedron, all faces of those forms can be 
similarly expressed. 
118. Given and X 3 to determine and \ v and also and 
X 9 in terms of the former. ^ , TTT . N 
From the spherical triangle 0,f0, (fig. 40*, Plate IV.*), we 
have bv the formula of spherical trigonometry, 
~ ” - e ' ““ n n n * "Ob / , 
at 
cos f0,= COS C/1, cos Oj/+sin 0,0, sin 0,f cos. f 0,0, ; 
but the spherical angle f0,0, is measured by the arc gO. 
the equator. „ , . ,, 
Hence, substituting the values of these arcs given in the 
previous section, we have 
cos_Pi =cos 90° cos _p 3 + sin 90 siny> 3 cos X 3 
== sin y> 3 cos X 3 . 
Again, in the spherical triangle fgC z , we have 
si nfg __ sinfC s g 
sin / Co sin fgC 3 ’ 
