:me. h. j. beooke on the geometeical isomoephism of ceystals. 
Magnitude of Indices . — It has also been suggested that the numerical values of the 
indices of secondary faces should in some degree influence the choice of the form to be 
regarded as the primary. For as it has been observed that in the cubic system the 
highest numerical index is seldom greater than 6, it has been inferred as a sort of law 
that aU indices ought to be expressible in low numbers, and that any form which allows 
the other forms o^ the crystal to be denoted by the lowest numbers, is on that account 
entitled to a preference as a primary form. 
But on looking through the lists of indices in the new edition of William Phillips’s 
‘Mineralogy,’ several instances occur of indices as high as 23, and there does not appear 
theoretically to be any reason why still higher numbers might not be so employed, as 
they occasionally are in the accompanying Tables, and in a recent paper on Quartz by 
M. Descloizeaux, in which indices of observed direct faces are as high as 32, and of 
direct faces corresponding to observed inverse faces as high as 80. 
The terms direct and inverse faces relate only to rhombohedral crystals, and may be 
explained as follows : — 
Let h k k be the sjunbol of any dhect face in the quadrant 1 1 1, 1 0 0 of the zone 111 
10 0, and lei pgg be that of a face in the same zone lying on the opposite side of the 
principal axis of the crystal and making an equal angle with the face 111. Then pgq 
is termed an inverse face. 
It frequently happens that direct faces and the corresponding inverse faces occur in 
pairs upon the same ciystal. 
Sometimes, however, a face occm’s in a cUrect position without a corresponding inverse 
face, and sometimes an inverse face occurs without any corresponding direct face. In 
the cases where the mverse faces occm* alone, the symbols of the corresponding direct 
faces have been computed and entered in the accompanying Tables, E 1 and E 2, as 
those of dhect faces. 
But it is necessaiy to consider the subject of high indices rather more in detail. 
The indices of faces do not express absolute numbers high or low, but only the ratios 
of the numbers so employed. Thus the symbol of 1 0 2 is equivalent to, and denotes 
geometrically, the same face as 2 0 4, 3 0 6, 10 0 20, 99 0 198, or any other numbers in 
the same ratio. 
It is usual to express indices in theh lowest terms in order to avoid writing figures 
unnecessarily, and to save time and trouble in the calculations of indices and angles. 
But it has the disadvantage of masking, and thus keeping out of sight, the true relations 
among such indices as are now under consideration. 
Let it, for example, be proposed to ascertain what faces can occur in the pyramidal 
system between the faces 101 and 10 3. The face 10 2 first presents itself as the inter- 
mediate face. But let 101 be expressed by 2 0 2, and 10 3 by 2 0 6, and 2 0 3, 2 0 4, 
and 2 0 5 appear as intermediate faces, and present to the eye a relation which is not so 
immediately perceived if we state the numbers as 2 0 3, 1 0 2, 2 0 5. If the expressions 
101 and 10 3 become 3 0 3 and 3 0 9, the symbols 30 4, 30 5, 306 ( =1 0 2), 3 0 7, 3 0 8 
