106 
Mr. Whewell on calculating 
7. The right square Prism. 
24. In this case, besides the co-existent planes which we 
have in the last figure, we shall have those which arise from 
considering that the sides AB, AC are symmetrical, that is 
p and q are permutable. Here the symbol is r) 
this will give eight secondary faces. 
8 . The Cube. 
25. This differs from the last in having the edge in the 
direction z similar to those in x and y. Hence p,q, r may be 
permuted and the symbol is l^pi^q, r) which gives 24 se- 
condary faces.* 
There is no necessity to vary the sign of r, for the plane 
q; — r) is the same as ( — p ; — q\ '^)- 
§ 3. The regular Tetrahedron and Octahedron. 
26. In this and other cases where the figure is bounded by 
more than three planes we shall make three of the primary 
faces co-ordinate planes, and the remaining primary faces 
will be expressed by different symbols. Also the co-existent 
planes will be differently represented accordingly as they are 
on one angle or another, and we shall in each case have to 
determine the different forms which will thus occur. 
Let Ax yz, fig. 12, be a regular tetrahedron, and let A<r, 
Ay, A % be three co-ordinates. . >. 
♦ In some cases however, we have only half the number of faces which the law 
of symmetry would give. Thus in the case of the pentagonal dodecahedron derived 
from the cube, the law is (2, i, o); but the faces which occur are (2; i ; o) 
(i ; o; 2) (o; 2; i) which by the changes of sign become 12. The other 12 which 
arise from the symbols (1 ; 2 ; o) (2 ; o ; 1) (o ; i ; 2) are excluded. 
