ME. H. J. BEOOKE ON THE GEOMETEICAL ISOMOEPHISM OF CETSTALS. 
these faces to either of the others, and has thus established a sort of probability that the 
face so preferred will occur more frequently than the others. 
Is it, however, true that the others have not been observed, although the observations 
have not been recorded; or is it not equally probable that such faces have been frequently 
seen and measured, but from the nearness of their angles with other faces, to the angles 
of the face 10 2 with the same other faces, they have been taken as imperfect instances 
of the face 1 0 2 1 
If the three faces above referred to should occur on the same crystal they might 
doubtless be distinguished, but if they occur on different crystals there is only one natural 
test for ascertaining whether they are really different, or are only imperfect examples of 
the same face. 
This test is, to find whether the face on one of the crystals is common to any two zones, 
and whether the face on the other crystal is common to the same two zones. If the 
faces are common to the same two zones on both crystals, the faces are similar, and if 
they are not, they are different faces. But from the occurrence of crystals generally in 
only small fragments, it does not often happen that any of these are sufficiently perfect 
to present clearly the faces of any two required zones, and hence the difficulty of ascer- 
taining with certainty the identity or otherwise of such nearly corresponding faces. 
But although there may be no theoretical limit to the magnitude of indices, there is a 
practical limit, beyond which faces denoted by high mdices would not be distinguishable 
with certainty from each other. This limit occurs when the difference of the angles of 
any two faces with some other face is within the probable range of error of ordinary 
measui’ement, arising from imperfection of faces, or from imperfect observation. It is 
difiicult to state the probable amount of error which may be ascribed to these causes, but 
perhaps about half a degree to a degree may not be an improbable quantity. 
The following are the several angles between the face 0 01 and the first and last ten 
faces in the preceding list : — 
20 
0 
21 = 
O 
43 
11 
20 
0 
26 = 
= 37 
/ 
9 
20 
0 
51 = 
O 
21 
/ 
8 
20 
0 
56 = 
0 
19 
23 
20 
0 
22 = 
41 
51 
20 
0 
27 = 
= 36 
7 
20 
0 
52 = 
20 
45 
20 
0 
57 = 
19 
4 
20 
0 
23 = 
40 
35 
20 
0 
28 = 
= 35 
8 
20 
0 
53 = 
20 
24 
20 
0 
58 = 
18 
46 
20 
0 
24 = 
39 
23 
20 
0 
29 = 
= 34 
12 
20 
0 
54 = 
20 
3 
20 
0 
59 = 
18 
28 
20 
0 
25 = 
38 
15 
20 
0 
30 = 
= 33 
18 
20 
0 
55 = 
19 
42 
20 
0 
60 = 
18 
11 
It appears from the small differences in the angles in this statement, that some of the 
adjacent faces in the preceding list might not be distinguishable from each other by 
ordinary measurement, and that the zone test already alluded to might here become 
necessary. 
But besides these possible sources of error in our crystallographic investigations, 
another element of disagreement occurs in practical crystallography. 
The ordinary method of examining a crystal is, first, to determine, from the character 
and relative positions of its faces, the system to which it belongs. 
With regard to the cubic, pyramidal, and rhombohedral systems, this process is 
