34 ME. H. J. BEOOKE ON THE GEOIVIETEICAL ISOMOEPHISM OE CETSTAXS. 
appear to be those of possible intermediate faces. And when higher multiples are 
employed, the number of possible intermediate faces is increased. 
If the multipher is 20 the original symbols become 20 0 20 and 20 0 60, showing 
thirty-nine possible intermediate faces between 101 and 1 0 3, as in the following list: — 
20 0 20 =10 1 
20 0 21 
20 0 22 = 10 0 11 
20 0 23 
20 0 24 = 10 0 12 = 5 0 6 
20 0 25 = 4 0 5 
20 0 26 = 10 0 13 
20 0 27 
20 0 28 = 10 0 14 = 5 0 7 
20 0 29 
20 0 30 = 10 0 15 = 406 = 203 
20 0 31 
20 0 32 = 10 0 16 = 5 0 8 
20 0 33 
20 0 34 = 10 0 17 
20 0 35 = 4 07 
20 0 36 = 10 0 18 = 5 0 9 
20 0 37 
20 0 38 = 10 0 19 
20 0 39 
20 0 40 = 10 0 20 = 408 = 204=102 
20 0 41 
20 0 42 = 10 0 21 
20 0 43 
20 0 44 = 10 0 22 = 5 0 11 
20 0 45 = 4 0 9 
20 0 46 = 10 0 23 
20 0 47 
20 0 48 = 10 0 24 = 5 0 12 
20 0 49 
20 0 50 =100 25 = 40 10 = 205 
20 0 51 
20 0 52 = 10 0 26 = 5 0 13 
20 0 53 
20 0 54 = 10 0 27 
20 0 55 = 4 0 11 
20 0 56 = 10 0 28 = 5 0 14 
20 0 57 
20 0 58 = 10 0 29 
20 0 59 
20 0 60 = 10 0 30 = 5 0 15 = 1 0 3 
It is apparent from this list, that when the number of faces in a zone increases, one 
or more of the indices by which the additional faces can be expressed must become 
larger. 
It is equally clear, that when the number of faces in a zone increases, the angles 
between adjacent faces must become less, and hence when the angle between any two 
adjacent faces in a zone becomes less, one or more of the indices requii’ed to denote one 
or both of such faces must become larger. To limit therefore the magnitude of indices 
would virtually be to limit the number of possible faces in a zone. 
We have before shown that the indices of faces are the reciprocals of the proportions 
of the primary edges conceived to be cut away by such faces, and in order to pro'^ide an 
exact expression of these proportions, in the cases of high indices, we have only to imagine 
the primary edges divided into a sufficiently large number of equal parts to allow of such 
an exact expression. 
It is evident that the symbols in the preceding list accurately denote separate faces. 
But it may be asked, are all these faces equally possible, and if so, are they equally pro- 
bable ; or is 20 0 40 more probable than 20 0 39, because the indices 20 0 40 can be 
divided by 20 without a remainder, and because those of 20 0 39 cannot! There does 
not appear to be any theoretical reason why any one of these faces should be more pro- 
bable than any other. 
But it may be said that the face 20 0 40 has been frequently observed and expressed 
by the symbol 10 2, while there is no pubhshed record of the faces 20 0 39 or 20 0 41 
having ever been noticed, and that nature has therefore shown a preference of one of 
