manner by (P+oo)m, inasmuch as the latter are limits of the series of the pyramids themselves.

According to the derivation employed in the crystallographic method of Professor Mohs, a scalene six-sided pyramid is obtained from a rhombohedron, by lengthening the axis of the latter on both sides to an indefinite but equal length, and joining the terminal points of the axis thus determined, and the lateral angles of the rhombohedron, by straight lines, as in Fig. 16. The number m expresses the ratio of the lengthened axis A'X', to the original one A X. Two forms, a rhombohedron, and a scalene six-sided pyramid thus connected with each other, are considered as co-ordinate members, or such as belong together in their respective series.

The edges of combination between s and u are parallel to the lateral edges of R; so are those between s and b. Both the pyramids, therefore, belong to R, and their general sign, n being equal = 0, becomes (P)m, where the exponent m is still to be determined.

The situation of u is exactly determined by the parallelism of the edges of combination between a and u, and those between u and e. Suppose in Fig. 19., AEB, ABD, ADC to be three faces of the isosceles six-sided pyramid P, ABKC one of the faces of the rhombohedron R. If a face of the scalene pyramid in question passes through the point B, its intersection with ABKC will coincide with the edge BK, the latter being the edge of combination between the faces of R and P+∞; but the line BN, its intersection with ABD, will bisect the terminal edge AD of the pyramid P in the point N, because B'BNCC' denotes the direction of one of the faces of R +∞ (e), passing through the point B. The line KI, which joins the lower angle K of the rhombohedron R, with the angle of combination N, determines the situation of I, the apex of the required pyramid, one of whose faces, therefore, is the triangle IBK. The axis of the derived pyramid is = 2 AI + § AG, AG being = § of the axis of R, and AI the prolongation of the axis on one side of the rhombohedron, analogous to AA' in Fig. 18. Now, IA+AH: HN = IA + AG : GK,

and a being the axis of R,

IA+ a:}=IA+a: 1.

Hence IA=a, and the axis of the pyramid, being=a+a, isa. The exponent m is therefore, and the original sign for the pyramid = (P), which, on account of the peculiar character of the combinations, still must undergo a farther modification.

The pyramid b is determined by the parallelism of the edges of combination between s and b, and between b and M. In Fig. 20., as in the preceding, ABKC is one of the faces of R, but A'EB, A'BD, and A'DC, are three faces of P+1, and not of P, because in this consists the difference between the two The axis of the derived pyramid is equal to 2 IA + & In order to find IA, we have

cases.

AG.

Hence IA = a, and the axis of the pyramid = a. The original sign of the pyramid becomes therefore (P)}.

The signs (P) and (P)3, although they in general denote the direction of the faces, yet do not suffice for expressing that mode in which they are contained in the combinations of the species. According to the method of Professor Mohs, the sign of a dirhombohedron in general is 2 (R+n), that of s is 2 (R); in a like manner the dipyramids are in general designated by 2 ((P+n)m). Supposing, in the developed combination, all the faces of the pyramids to appear, these signs would become, 2 ((P)3) and 2 ((P)3). But there are only the alternating faces to be observed in the combinations, and the way to denote the pyramids, including the situation of their faces to the right

or the left of the faces of R, will therefore be :

1 2 ((P) :)

r

2

as referring to the figures. The addition of the

letters r (right), and 1 (left), is required for distinguishing the combinations occurring in apatite, from those which are to be met with in quartz, where, in different individuals, we have to 1. 2 ((P)3)

r

express by

2 ((P) 3)

r

(Fig. 3.) and

(Fig. 4.), that

in these combinations contiguous to both the apices, only the right, or only the left faces of the scalene six-sided pyramids can be observed. The inclination of u to M is 150° 40.5', that of b to M=157° 26; this inclination being equal to the sum of one-half of the angle at the lateral edge of (P + n)m, and 90°*.

As to the twelve-sided prisms, it is evident, from the horizontal edges of combination between u and c, that the latter is the limit of that very same series of which the former is a member. Since it does not appear with the full number of its faces, but only with those which, considered from one extremity of the crystal, appears on the left, from the other on the right of the

faces of R, its representative sign will be 1 (P + ∞ )§.

r 2

For the want of appropriate edges of combination, I have been obliged to resort to immediate measurement for ascertaining the position of the faces, marked ƒ, and the law, by which the form produced by these faces depends upon the fundamental rhombohedron R. The prism in question is the limit of that series of six-sided pyramids, whose derivative exponent is the number 3. The sign of these faces as they appear in the combination, to the right of R on the one side, and to its left r (P + ∞ )3 ̧ I

on the other, will therefore be

2

The angles of the transverse section of the two twelve-sided prisms are the following:

The lateral edge z Fig. 16., is obtained by the formula,

a being the axis of the rhombohedron, and m the exponent or number of deriva

tion upon which the pyramid depends.

The transverse sections of the two prisms are therefore equal to each other as to the measure of their angles, but they differ n the relative situation of their more acute and more obtuse edges. In the combination, Fig. 16., which, besides R + ∞ (e), and P+∞ (M), contains the faces of (P +∞)§ (c) situated to the left, and those of (P +∞)3 (f) situated to the right of the faces of R+∞, the transverse section is a figure of twenty-four sides, the angles of which are alternately 169° 6′ 24′′, and 160° 53' 36".

If in Fig. 18. the more acute terminal edge of the scalene six-sided pyramid is called x, the more obtuse one y, the lateral edge z; and we suppose the axis of the rhombohedron to become infinite, the pyramid is transformed into a twelve-sided prism, of which the alternating angles are equal. The edge y becomes vertical, and appears in the combinations contiguous to the faces of R+∞ (e), the edge z also becomes vertical, and appears contiguous to the faces of P+∞ (M), exactly as is mentioned in the preceding Table. The reversed equality of the angles in two different prisms, as given in this Table, depends, therefore, upon that of the geometrical expressions for these two edges.

The general expression for the edge y is

a being the axis of the rhombohedron, to which the pyramid belongs, and m the number of derivation, or that which expresses the ratio between the axes of the two forms.

If now, a being infinite, cos y of (P+∞)m is supposed = cos z' of (P +∞)m', for a certain m in the first, and another m' in the second expression, we obtain

If, in these formulæ instead of m, we substitute §, the value of m' becomes 3, and vice versa; so that, in fact, the two numbers of derivation § and 3 will yield twelve-sided prisms, whose alternating angles are inversely equal to each other.

៖

GOIN

The substitution of, the number of derivation of the other pyramid, in these formulæ, instead of m, would make m′ =2. q{}

៖

Numerous very interesting results might still be obtained from a further continued comparison of the forms occurring in the crystals of apatite, and in those of other species which belong to the rhombohedral system. Among these I shall only mention, that the scalene six-sided pyramids, derived according to the numbers and 3, are far less generally to be met with in nature than those dependent upon the numbers 3and 2. Yet they have already been observed in several species, for instance (P)? in rhombohedral quartz, being the pyramid noted a in Haüy's Figures, and in Figs. 3. and 4. of the present paper, (P)§, and (P+1); in calcareous-spar, &c. whilst the peculiar character of the combinations of apatite, in as far as our present knowledge reaches, is quite unparalleled in the Rhombohedral System, in which it stands as isolated, and as remarkable as the series of crystallisation of rhombohedral quartz.

There exists a striking analogy between the forms of Apatite in the rhombohedral, and those of Tungstate of Lime, (Pyramidal Scheelium-baryte) in the pyramidal system. A more detailed examination of this analogy, how interesting soever it might prove, is a subject which requires so many particulars, that it will be better to defer it to some future occasion.

ART. XXIV.-Account of some remarkable and newly discovered Properties of the Suboxide of Platina, the Oxide of the Sulphuret, and the Metallic Powder of Platina. By Prof. DÖBEREINER of Jena *.

MR EDMUND DAVY had shewn, in 1820, that sulphate of platina, if boiled in alcohol, and subsequently digested in am

• Translated and abridged from the original paper published in Gilbert's Annalen der Physik, which Professor Gilbert has been so kind as to transmit to us previous to its publication in his valuable Journal.-ED.