322 Scientific Intelligence. 



with^each other. From the above data we have the volume of 

 the tetrahedron v, from the formula 



V = -J abc y'sin s . sin [s — a] . sin [s — ft] . sin [s — y\ . 



where s represents the half sum of the angles a, /3, y. The vol- 

 ume of the entire, fundamental form would consequently be eight 

 times as large, or in rectangular coordinates, V = ^abc. The 

 contents of the corresponding ellipsoid is, in rectangular coordi- 

 nates, where the coordinate axes become the axes of the ellipsoid, 

 = f n abc. 



For the volumes of the various crystal systems we have, if we 

 denote the quantity under the radical by A, the following for- 

 mulas : 



I. Regular ; CV =^7t (where a — b = c = 1). 



II. Tetragonal and Hexagonal ; CV = f nc (for a = 1), = ^ 7ta 2 

 (for c = 1). 



III. Rhombic; CV = § nac (for ft = 1), =^7rbc (a =. X)^ — ^nab 



IV. Monoclinic and Tri clinic ; 



CV=7r|ac A /A (5=1), etc. 



For the regular system the ellipsoid is a sphere, for the tetra- 

 gonal and hexagonal systems an ellipsoid of revolution, and for 

 the others an ellipsoid whose semiaxes are x 9 y, z. 



If the crystal-volume so computed were equal to the actual 

 volume of the smallest crystal, or to that of the molecule, then it 

 would only be necessary to multiply this value by the specific 

 gravity of the substance under consideration in order to obtain 

 its molecular weight in terms of water as the unit. The value of 

 the crystal-volume as computed is, however, only proportional to 

 the actual volume of the molecule, and hence the product, d . CV, 

 represents only a value proportional to the molecular or atomic 

 weight, taken with reference to water or hydrogen as the case 

 may be, and this value is to be multiplied or divided by some 

 number in order to give the molecular or atomic weight. Since, 

 however, this number is not known, we are not able to use the 

 computed value d . CV. We can use it, however, as soon as we 

 consider the entire eutropic series and apply the law of eutropy 

 to the values thus found. 



It is therefore evident, that in the case of eutropic crystals, the 

 crystal-volumes must form a series, such that with increasing 

 atomic or molecular weight they either decrease, or what is more 

 probable, increase. In like manner the products d.CV, that is, 

 the crystal-weights, must stand in accurate ratio to each other. 



We ask further, are then the computed crystal-volumes of all 

 members of a series mutually equivalent? This also must be 

 answered in the negative, for it must not be forgotten that in 

 each substance the lengths of the coordinate axes are taken with 

 reference to a new unit of length, since in each case we put one 

 axis equal to unity. We ought, however, to refer the crystals of 

 the various members to the same unit of length, and only then 



