ON DIMORPHOUS BODIES. 171 
or that in these forms the molecules are nearest to each other 
in the following order :— 
Reg. Octohedron and Cube. 
Rhombic Octohedron ?. 
Rt. Rhombic Prism. 
Oblique Rhombic Prism ?. 
Rhomboid. 
The hardness of the several forms seems to follow a similar 
order, the denser of two forms being also the harder. This is 
certainly the case with the diamond and the arragonitic forms of 
carbonate of lime and baryto calcite, but the observations we 
possess on this point are still too few in number, and made, in 
general, with too little attention to minute accuracy*, to justify 
us in founding any general conclusion upon them. 
20. It will be observed that the several members of each 
group in the above table are represented by analogous formule 
with a substitution in each of one element only,—a metal. The 
first group, with one exception, is represented by the general 
formula RC or RO+CO,, and the fourth group by RS +7H Isomor- 
or RO+RO,+7HO in which not only the entire sum of the Phous 
i ct A 5 Toups, pro- 
negative and positive equivalents is equal, but the sum of those ably ak. a 
ineach member of the formulz is also equal. Thus in the first of morphous. 
these formulese RO+ CO, the negative are to the positive equiva- 
lents as 3 : 2, and in the two parts RO and CO, they areas 1:1 
and 2:1. This is the case with all the neutral carbonates of 
Protoxides, whether isomorphous or not. In the second for- 
_ * M. Frankenheim has observed in regard to the hardness of crystallized 
bodies, native and artificial, that three orders of differences are to be observed : 
1° On the same line in opposite directions. 
2° On the same face in diferent lines. 
3° On different faces of the same crystal. 
He finds that two directions or faces of the same crystallographic value have 
always the same hardness, and that isomorphous bodies very different in abso- 
lute have similar relative hardnesses. This is the case, for example, in regard 
to nitrate of soda and calv spar, the absolute hardnesses of which are so unlike. 
These orders of differences he found to be intimately connected with the 
natural joints of the crystals, so that the hardness is least. 
1° Inrelation to different faces; on the faces of the joints themselves. 
2° On each face in the line perpendicular to the intersection which the prin- 
cipal joint would give of that face. 
__ 3° On the same perpendicular, in a direction from the obtuse to the acute 
dihedral angle of the intersection.—Frankenheim Traité sur la cohesion des 
corps. Extract Biblioth. Univ., June 1886. 
By considerations drawn from the relations ‘of the pelar forces, supposed to 
reside in the crystallographic axes of crystallized bodies, M. Voltz has endea- 
voured to show that the hardness must vary on different faces and in different 
directions, and according to certain laws (/’Jnstitut., 9th August, 1834). : 
