262 
Mr Levy on the Modes of Notation 
]y to the primitive. It is obvious a priori, from the symmetry 
of a rhomboid, that two equal dodecaedrons, differing in posi- 
tion, and the principal sections of which are inclined at an angle 
of 60°, may be derived from the same primitive form, and the 
two last formulas determine the indices of one of them, when 
those of the other are known. If it were required, for instance, 
to determine the indices of the dodecaedron, similar to the me- 
tastatic of carbonate of lime, but differing as to its position with 
respect to the primitive, it would be sufficient to substitute in 
the two last formulae for x, y, z, the indices of the metastatic, 
which are x .== 1, y = o, z 
2, and then the values of and 
Sr 
x . . 2 4 
— will be found respectively equal to — and — , and consequent- 
'll ^ 
ly the sign of the required dodecaedron will be (6 1 b 2 b 5 ), or, 
according to Haiiy’s notation, (E^ B 1 D 2 ). This modification 
is one of those he has described, and he mentions other instances 
of more equal dodecaedrons produced by two different laws of de- 
crements, and I have had occasion to observe several others. 
There is, however, one case in which the values and are 
found to be respectively ^ and — ; and consequently, in that 
case, the two laws of decrements are the same, and the two dode- 
caedrons are not only equal, but their positions are the same, 
with respect to the primitive. This takes place when z . = 1, 
x y 
or y — Z — X- — y y that is to say, when the dodecaedron is com- 
posed of isosceles triangular planes. This last remark proves, 
that there is an infinite number of dodecaedrons with isosceles 
triangular planes, produced by intermediary decrements, and 
that for all these there exists between their indices the following 
relation, 2t / = z + x. 
The last point to be considered, is the determination of the 
indices of a dodecaedron, when two of its incidences are known. 
Those which generally are most readily measured, are desig- 
nated by ( i ' i) and (i : i). Three of the preceding formulae de- 
termine immediately the indices of the dodecaedron, when they 
