ns 
the angles of crystals, 
and P' Q' R' will be co-existent planes ; and the condition of 
their co-existence is included in the preceding symbol. 
The quantities a, b, c are z.s na^n 6, n r, that is as A x, y z 
and Ay. Or, referring to the octahedron in fig. 13, they 
are as FH, FE, and FD. 
The square Octahedron. 
32. When EFHK, fig. 13, is a square, hx^yz will be 
equal, and the solid angles aty and z will be symmetrical to 
those at A and x, and will be similarly affected. Hence for 
a pkine at A there will be co-existent planes at y and 2;. 
Prop. To find the symbols of co-existent planes in this case, 
If we takers P', z Q', %R',=y/, y (/', y r',= Kp, A^, A r re- 
spectively, we shall, as in last article, find the equation of the 
planes p' q' r\ P' Q' R' to be 
and since/) = q=z-^, r = -i, these are equivalent to 
i<I-r)i+qi+{q-p)T = ^^ 
(q-r)i+(q-p)^+q-^=^ 
Hence with a plane {p’, q', r) we have co-existent planes 
(q — r; q; q — p) and (q — r; q—p; q). 
But we have also a co-existent plane (/>; r; q) and therefore 
also , (r — q;r;r — />) and (r — q;r — /> ; ^) 
Hence in the square octahedron we have co-existent planes 
which may be included in this symbol 
{iPi q> r]ip-- p — r,p — q)(q — r; q, q—p)(r—q;r,r—p}^ 
All which are implied in {p[ q r). 
MDCCCXXV. Q 
