90 
Mr. Whewell on calculating 
Since therefore the direction of the plane P Q R is com- 
pletely determined by the three quantities h, k, /, we may re- 
present it by writing those three quantities thus (y; y ; y) 
or, if the equation be px -f rsr = we may represent 
the plane by the symbol {p; q; r). 
S. According to the law of symmetry which prevails in 
the production of crystalline forms, if one edge or face of the 
primary solid be modified in any manner, the other homo- 
logous edges and faces will be similarly modified. Hence, if 
one plane exist, other corresponding planes must also exist, 
and these we may call co-existent planes to the first. 
Thus if we have a plane PQR, Fig. 2, and if we take 
AP' =; AQ, and AQ' = AP, we must also have a plane 
P'Q'R: for the edges hz, Ay being perfectly similarly 
situated, if one of them be affected in any manner, the other 
must be similarly affected. Hence, if we have a plane 
[p; q; r), we must have one {q ; p ; r). The same is also 
true of % ; and by considering this in the same manner, it 
will be seen that the plane [p ; q; r) has the following co- 
existent planes 
{q;p;r) (r-,q;p) {p;r;q) (q;r-,p) (r;p;q). 
That is, there are all the permutations that can be made by 
altering the arrangement of the three quantities^, q, r; that 
the one which stands first in order being always the coefficient 
of X, the second that of y, and the third that of z. 
These six planes may be represented by a single symbol 
* We might represent the plane hy {h; k; 1), which shows more immediately the 
law of its formation ; but in all our subsequent calculations we have to use the re- 
ciprocals, and hence our formulx are simplified by using the symbol {p; r) where 
p, q, r are the coefficients of the equation. 
