375] 
52S> 
375. 
NOTES ON POLYHEDRA. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. vn. (1866), 
pp. 304—316.] 
Axial Properties. Article 1 to 18. 
1. A polyhedron may have a q-axis, viz. a line about which if it is made to 
27,- 
rotate through an angle =— (but not through any sub-multiple of this angle), it will 
iZ 
occupy the same portion of space. It is then clear that when the rotation is repeated 
any number of times the body will still occupy the same portion of space; or if © 
2? t 
denote the rotation through the angle —, then we have the rotations 1, ©, © 2 ,... ®^ -1 , 
and finally ©« = 1, that is, when the rotation is q-times repeated, the body will resume 
its original position. Similarly for any number of axes (©« = 1, ©'s' = 1,..., where the 
indices q, q',... may be the same or different) we have the rotations 1, ©, © 2 ,... @® -1 , 
©', @' 2 , ... ©'s' -1 ,...; and if ©, ©',... be the entire system of the axes of the body, 
these rotations will form a group. The rotations in question are in fact the entire series 
of those which leave unaltered the portion of space occupied by the body, and since 
any two rotations combine together into a single rotation, any two of the rotations 
in question must combine together into some one of these rotations, that is, the 
rotations in question form a group. Some analytical consequences of this theorem will 
be obtained in the sequel. 
2. The number of axes may be denoted by Si and the number of rotations by 
1 + ^ (q - 1); we may say that Si is the number, and 1 + S (q -1) the efficiency or 
weight, of the axes. 
C. V. 
67