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the angles of crystals. 
P, Q are any number whose sum is 2. Thus (1, 1, — 1) 
(2, o, — 1) are truncating faces. 
(This rhomboid truncated by these two planes occurs in 
Hauy's Chaiix Carbonatee Progressive. Fig. 41-) 
The plane thus determined will always be parallel to the 
intersection of the two planes ; but in order that it may trun- 
cate the edge, it must meet both of them on the really exist- 
ing part of each plane. This condition is easily introduced 
in each particular case. 
16 . In order to express, by means of the symbols already 
introduced, any crystal whatever, we may write down the 
symbols of the faces by which it is bounded ; indicating by 
the punctuation the permutations which are allowed. It will 
be convenient also to mark the number of the faces which 
arise from these permutations. In the rhomboid, when all 
the three indices are different, this number will be six. When 
two are alike, it it will be three. Thus (6) (^, 9, r) may indi- 
cate that the crystal has six faces arising from the law ex- 
pressed by (^, 9, r) and ( 3 ) (g, p, r) may represent a crystal 
with three faces arising from the law 0’ which is 
what would, according to Hauy, be called a decrement on an 
angle at the summit. 
It often happens that faces in a crystal are repeated ; that 
is, that there are faces parallel to one another, one of 'which 
may be considered as a repetition of the other. In that case 
we may distinguish them by placing a 2 before them as a 
multiplier. Thus 2 ( 3 ) (/>, p, r) indicates a rhomboid pro- 
duced by repeating each of the three faces represented by 
(p, py r). This is in fact the mode in which a rhomboid is 
always produced. In the same manner 2 ( 6 ) (p, 9, r) is the 
