108 Mr. Whewell on calculating 
It being understood that in each parenthesis the indices which 
are separated by commas may undergo any permutation. 
The first symbol [p, q, r) gives 6 planes, and the three 
others also 6 each, making in all 24. 
If the primary form be known to be a regular tetrahedron, 
it is evident that the first symbol [p, q, r) must be understood 
as implying also the rest. But in order to express all the 
planes we may include them in one symbol thus 
{iP<q>r){p,p — q,p — r)Sic.^ 
the &c. implying the coexistent planes. 
27. Prop. To determine the symbol of the planes which 
truncate the edges of a tetrahedron. 
The plane truncating the edge a? is (o ; q, r) : and hence by 
last article the general symbol includes the planes 
[o,q,r), {q,q,q — r), (r, r— .g,r) 
which gives 12 planes. We omit (o, — q, — r), which is 
identical with (p,q,r). 
If ^ = r the planes are expressed by (o, q, q), which gives 
3 planes ; but in order to truncate the six edges, each is 
used twice, and the symbol is 2 (3) (o, q, q). 
The regular octahedron is bounded by the same 4 planes as 
the tetrahedron, each being used twice ; and its symbol is 
2 (4) { (1, 0,0) (i, 1 , i)| . 
Its edges are also parallel to the edges of the tetrahedron, 
each being used twice. And any plane which can be deduced 
from the octahedron, may with equal simplicity be deduced 
from the tetrahedron. 
28. Prop. In the regular tetrahedron to find the angle 
contained by planes (o, 1, 1). 
