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doubt that in such a case the water would soon regain its level; for the ocean at the equator being heavier than at the poles by the weight of a layer 4 feet in thickness, it would sink at the former place and rise at the latter until equilibrium was restored, producing, of course, a very slight displacement of the bottom-waters towards the poles. It will be observed, however, that restoration of level in this case takes place by a simple yielding, as it were, of the entire mass of the ocean without displacement of the molecules of the water over each other to any great extent. In the case of a slope produced by difference of temperature, however, the raised portion of the ocean is not heavier but lighter than the depressed portion, and consequently has no tendency to sink. Any movement which the ocean as a mass makes in order to regain equilibrium tends, as we have seen, rather to increase the difference of level than to reduce it. Restoration of level can only be produced by the forces which are in operation in the wedge-shaped mass WC W', constituting the slope itself. But it will be observed by a glance at the figure that, in order to the restoration of level, a large portion of the water W W' at the equator will require to flow to C, the pole.

According to the general vertical oceanic circulation theory, pressure from behind is not one of the forces employed in the production of the flow from the equator to the poles. This is evident; for there can be no pressure from behind acting on the water if there be no slope existing between the equator and the poles. Dr. Carpenter not only denies the actual existence of a slope, but denies the necessity for its existence. But to deny the existence of a slope is to deny the existence of pressure, and to deny the necessity for a slope is to deny the necessity for pressure. That in Dr. Carpenter's theory the surface-water is supposed to be drawn from the equator to the poles, and not pressed forward by a force from behind, is further evident from the fact that he maintains that the force employed is not vis a tergo but vis a fronte (Proc. Roy. Geog. Soc. Jan. 9, 1871, § 29). [To be continued.]

XV. On Quartz, Ice, and Karstenite. By W. H. MILLER, M.A., F.R.S., Professor of Mineralogy in the University of Cambridge*.

Quartz.

AMONG the minerals presented to the University by H. W

Elphinstone, Esq., are two crystals of quartz associated with chlorite, apparently from the same, but unknown, locality.

* Communicated by the Author.

Each of these crystals exhibits one face of a rhombohedron, having angles which differ too widely from those of the forms described by Des Cloizeaux in his Manuel de Minéralogie to admit of identification with any of them, and therefore has probably never been observed before.

The larger of the two crystals, besides the supposed new face, which will be denoted by the letter 5, has the forms 2 II, 100, 1 22, 8īī, 10īī, a 142, a 41 2. The faces 5, 100 are rather uneven, the bisection of the images of the bright signal being uncertain to the extent of about 2' in the former and rather less in the latter. Three observations of the angle between these faces gave 30° 23′5, 30° 23′5, 30° 24′ respectively.

The other crystal has the forms 2 II, 100, Ĩ 22, a 412, a 41 2 in addition to 5. This last face is very even and bright; but 100 is rather imperfect. The observed angle between these faces lies between 30° 22'2 and 30° 28'4.

Of the faces given by Des Cloizeaux, those which most nearly approach the position of 5 make with 100 angles of 29° 26', 30° 4′, 30° 44′, having for their symbols 11 44, 833, 1355 respectively. In order to obtain more probable values of the indices of 5, let us suppose the angle between 100 and 5 to be 30° 24', hk k the symbol of 5, and D, T the angles which the axis of the rhombohedron makes with normals to the faces 100, hkk. Then, since D=51° 47' and T-D=30° 24', we have T=82° 11'.

But

h-k tan T
tan D

=

h+2k

=5.7356.

The converging fractions approximating to this number are:-

5 6 17 23 86

I'I' 3' 4' 15'

The first two fractions give the faces 11 44, 1355 already noticed; the third a face 37 14 14, making an angle of 30° 18' with 100, and therefore not very probable; the fourth a face making with 100 an angle of 30° 25', which, taking into account the imperfections of the faces of the crystal, agrees sufficiently well with the observations. The resulting symbol is

50 19 19.

40

יד

The fraction obtained by adding the numerators and denominators of the third and fourth fractions, leads to the symbol 29 11 11. The face of which this is the symbol makes with 100 an angle of 30° 22′, and is therefore hardly so probable as

50 19 19, notwithstanding the lower values of the indices of the former symbol.

Ice.

In a memoir by Franz Leydolt, entitled "Beiträge zur Kenntniss der Krystallform und der Bildungsart des Eises," it is asserted that ice has no cleavage (Sitzungsberichte der mathem.naturw. Cl. der kais. Akad der Wissensch. Band vii. Abth. 2, p. 477). A good many years previously I had seen some plates of ice broken which exhibited a separation parallel to the surfaces of the ice so perfectly like cleavage, that I never hesitated to publish the statement that ice has a cleavage parallel to the faces of the form 111. A considerable time elapsed after the appearance of Leydolt's paper before an opportunity of making further observations presented itself. When at last I obtained some thick plates of newly formed ice, I was unable to procure a trace of cleavage by the application of knife, chisel, or point in a direction parallel to their bounding planes. On throwing one of the plates on the hard frozen ground it broke across, exhibiting in the fracture two planes normal to the natural faces of the plate, and apparently (for I had not at hand the means of measuring the angle they made with one onother) parallel to two adjacent faces of a regular six-sided prism, looking like very perfect cleavages, and affording by reflection distinct images of surrounding objects; but I was unable to obtain a trace of cleavage in planes parallel to either of those revealed by fracture. It is therefore obvious that the separations, as well parallel as normal to the surfaces of the plates of ice, were due to the existence of faces of union and not to true cleavage. The latter planes are probably those of the six-sided prism 101; for some crystals of ice examined by A. E. Nordenskjöld were combinations of the simple forms 111, 321, κ210, x513, 101. The angles which normals to the faces of these forms make with a normal to 111 are approximately :

111, 321=38° 57'; 111, 210=58° 15';

111, 513=81° 31'; 111, 10I=90°.

(Poggendorff's Annalen, vol. cxiv. p. 612.)

Karstenite.

A small cavity in the interior of a mass of Karstenite (CaO SO9) from Lüneburg was found to be traversed by several slender crystals attached at both ends to the walls of the cavity. These crystals exhibit some simple forms hitherto undescribed, and several of the forms first observed by Hessenberg and described by him in the 10th Number of his "Mineralogische Notizen,"

published in the Abhandlungen der Senckenbergischen Naturforschenden Gesellschaft, Frankfurt a. M. vol. viii.

Let the symbols 100, 010 denote faces normal to the lines bisecting the obtuse and acute angles between the optic axes, 001 a face parallel to the plane of the optic axes, and 1 10 the face which, according to the observations of Hessenberg, makes with 100 an angle of 48° 15', the angle between two planes being measured by the angle between normals to them drawn from any point in the interior of the crystal.

The crystals from Lüneburg exhibited faces of the forms 100, 010, 110, 210, 310, 510, 540, 320, 410, 430, 150, 520, 530, of which the 4th... 9th were first observed by Hessenberg. The last four appear to be new. The faces of all these forms give very indistinct reflections, with a single exception in one of the crystals a face of the form 110 was very perfect; the observed values of the angles it made with the cleavage 100 were:-131°, 45', 442, 44-55, 445, 447; mean 131° 44'-6. Hence 100, 110=48° 15'4. The angles between the different faces and the face 100, taking Hessenberg's value of the angle 100, 110, are :—

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When light is refracted through the prism bounded by the planes 100, 110, the least-refracted ray is polarized in a plane parallel to 001. The indices of the light in this plane are:

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In water one optic axis seen through the faces 110, 110,

and the other seen through the faces I10, 110, appeared to make with one another an angle of 36° 22'. Hence the direction of either optic axis within the crystal makes an angle of 22° 1' with a normal to the face 0 1 0.

In crystals from Berchtesgaden, Hessenberg searched in vain for a face of the form 101, for the existence of which I considered that I had had satisfactory evidence. I therefore reexamined the crystal in which I supposed it to be visible. Using a spot of sunlight reflected from a plane mirror as the bright signal, the crystal being adjusted so that the intersection of the faces 111, 111 was parallel to the axis of the instrument, an image of the spot of sunlight was seen as if reflected from a face making equal angles with the faces 111, 111; but on making the crystal revolve round the axis of the branch of the holder parallel to the plane of the circle, the spot remained immovable. Hence it is evident that the spot of light seen was not due to a single reflection, but to a reflection at each of two separations in the interior of the crystal parallel to the faces 001 and 100.

In several of the fragments of crystals from Berchtesgaden faint separations indicative of cleavages were observed, which, on measuring the angles they made with the face 100 with a position-micrometer, were found to be parallel to the faces of the form 110. It was not found possible to separate the crystal in the direction of this cleavage, on account of the superior facility of the other cleavages.

Some colourless crystals from Stassfurth, given me by M. Pisani, of Paris, had the faces of many of the simple forms striated to such an extent that it was extremely difficult to measure the angles they made with one another. By using for the bright signal a large beam of sunlight reflected from the mirror of a heliostat, I think I have ascertained the existence of the following forms, the last two being the least certain :—

100, 010, 001, 110, 210, 310, 320, 430, 510,

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