cos. S 
126 Mr. Whewell on calculating 
Substituting for A, B, A', B' in the formula, we have, since 
(p = 4/ = w 
, cos, y- 
pr 
p±q_ ^ q+r 
r p 
cos. (p- 
Ip + q q + r 
-f- 
r p 
cos. p 
V(> + I + — 4-2 cos. (p— cos. cos. p) (l -f I + 4- 2 cos. (p — 2 COS. p — 2 COS. 
2,pr -f 4- -I- — 2 (/ 4- r'^-^ pq 4- qr — p r) cos. p 
V ((/^ 4- ?)"^4- 2 r^—z {p q — r) r cos.ip)) {q 4- 4- z p'^—2(q r — p) pcos. p) 
And if we take any other opposite pairs of planes, (j!); r; q) 
{q; r; p) and (q ; p ; r){r; p ; q); or (r; q ; p)(r; p;q) and (p;r;q) 
(p; q\r)y we shall have the same value for 6. Hence this 
angle may be used as the characteristic of a pyramid pro- 
duced by any such law from a rhomboid : and consequently 
of a dodecahedron resulting from repeating the faces of the 
pyramid. It is employed in this manner by Bournon in cha- 
racterising the dodecahedrons of carbonate of lime. 
§ 10. Recapitulation. 
46. It may be useful to collect in one view the results of 
the foregoing investigations. If we take a solid angle of the 
primary form of a crystal for the origin, and the three edges 
for three co-ordinates, any secondary plane may be obtained 
by removing a pyramid, the edges of which consist of h, k, /, 
molecules respectively. If we make/>== ^,q = ^, r——^ 
the secondary plane may be represented by (p; q; r) which 
will express its position without determining its distance 
from the origin : p,q,r may be positive, o, or negative. 
By the law of symmetry with respect to the angles and edges 
of primary forms, if one secondary plane exist, certain others 
must also exist, which are hence called co-existent planes. 
Some of these are obtained by permuting the order of the 
letters in the symbol (/>, q, r ) ; and the instances where this 
