532 
NOTES ON POLYHEDRA. 
[375 
■octahedron form thus a system of rectangular axes common to all the polyhedra, and 
representing these axes (or say the summits of the corresponding rectangular spherical 
triangle) by X, Y, Z, we have a convenient system of coordinate axes to which to 
refer all the other axes of the polyhedron, viz. if P be the extremity (chosen at 
pleasure) of the axis in question, then the position of the axis may be determined 
by its distance PZ and azimuth XPZ (measured in the direction from X to F), or 
by its distances PX, PY, PZ, or say X, Y, Z from the three rectangular axes (we 
have, it is clear, cos X — sin dist. cos azim., cos F = sin dist. sin azim., cos Z= cos dist.). The 
2 7 r 
rotation angle of a ^-axis is = — (i.e. this i s tlie angle through which if the body 
be turned about the axis, it still occupies the same portion of space) and the half 
rotation angle is therefore = ^. Moreover if i, j, k are Sir W. R. Hamilton’s quaternion 
symbols, then the “ rotation symbol ” of the axis is 
cos — + sin - (i cos X + j cos F 4- k cos Z), 
qq 
the application of which will be presently explained. 
10. The angular coordinates of the different axes may be found by spherical 
trigonometry without much difficulty; and we are then able to form the following axial 
tables of the several polyhedra: the extremity of each axis is chosen in such manner 
that the distance PZ is not > 90°. 
Axial System of the Tetrahedron. 
Distances 
angle cos 
sin 
angle 
Azimuths 
cos 
sin 
cos X 
cos Y 
cos Z 
1 Rot. Symbols 
4 3-axes, | Rot. 
angle = 60°, cos = ^ 
sin = \ J3. 
54°44' 
33 
33 
33 
1 
ß 
33 
33 
33 
ß 
ß 
33 
33 
33 
45° 
135° 
225° 
315° 
i 
+ 71 
33 
33 
+ 33 
-* - = 
+ +11 
1 
+ ß 
~ 33 
“ 33 
+ 33 
1 
+ ß 
+ >j 
~ 33 
~ 33 
1 
+ —JZL 
s/3 
+ „ 
+ J> 
+ >5 
1(1 -t-i+j + le) 
è(l-i+j + k) 
i(l -i-i+A) 
i(! + i-j+k) 
3 2-axes, ^ Rot. 
angle =90°, cos = 0, sin = 1. 
0° 
1 
0 
* 
* 
* 
0 
0 
1 
k 
90° 
0 
1 
0° 
1 
0 
1 
0 
0 
i 
53 
33 
33 
90° 
0 
1 
0 
1 
0 
3