836 The American Naturalist. [September, 
of the contained metals and the crystal characters of the potassium, 
rubidium, and cæsium double sulphates of formula R,M(SO,),6H,O,5 
have been found by Muthmann' to hold also for the permanganates. 
Continuing his studies Tutton’ has made an equally exhaustive crys- 
tallographic study of the normal sulphates of the same alkali metals. 
The earlier determinations made on these substances seemed to be in 
conflict with the facts brought out by Tutton in studying the double 
sulphates, but after most exhaustive and precise observations with 
specially-devised apparatus, Tutton is able to show that the recorded 
observations on these salts are incorrect, and that the intermediate 
position crystallographically of rubidium is established for this series 
as well as the other. There isshown to be a progression corresponding 
to the increase of atomic weight of the contained metal as regards the 
axial ratio, the size of the interfacial angles, and the molecular volume. 
The differences in the magnitude of the analogous angles, seems, how- 
ever, to be less, the higher the symmetry, approaching, Tutton suggests 
the absolute identity requisite to isometric symmetry. The habit of 
the crystals seems to obey the same law. In a discussion of the rela- 
tive linear dimensions of the crystal elements of the Bravais-Sohnche 
space lattice, is communicated a simple method of determining these 
values which was suggested by Becke. Becke’s formule are: 
a= fav b= PV o= pev 
P3 ac a 
in which a,, b, and c, (X ¥ Z of Muthmann) are the relative dimensions 
of the crystal element in the direction of the correspondingly named 
crystal axes; a, b, and ¢ are the unity lengths of the crystal axes ; and 
V is the molecular volume. Tutton proposes to call the distances a, b, Co 
(Muthmann’s topische axen) distance ratios of the crystal elements, and, 
as they are only relative values, to make one equal to unity as in the 
case of, axial ratios. When these values are determined for the three 
sulphates, it is found that rubidium occupies the intermediate position, 
Tutton also finds that these salts follow the Bravais-Sobnche theory in 
that the planes of cleavage 4 (010) most perfect and (001) less perfect } 
are the planes in which the elementary parallelograms of the lattice 
system are respectively smallest and next smallest. 
The optical study consisted in the determination of the principal 
indices of refraction in prisms prepared with unusual care by the deli- 
5 See these notes. 
_ *Zeitsch. f. Kryst., xxii, p. 497. 
T Jour. Chem. Soc. London, 1894, pp. 628-717. 
