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50 19 19, notwithstanding the lower values of the indices of the former symbol.
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, x210, 513, 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, 101I=90°. (Poggendorff's Annalen, vol. cxiv. p. 612.)
A small cavity in the interior of a mass of Karstenite (CaO SO3) 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° 41.6. 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 :—
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:
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 01 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 king 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,
XVI. Measurements of the Polarization of the Light reflected by the Sky and by one or more Plates of Glass. EDWARD C. PICKERING, Boston, U.S.*
THE following observations, which will be published in full in the 'Proceedings of the American Academy of Arts and Sciences,' were conducted to test Fresnel's formula for the reflection of light. He showed that, if the light was polarized in the plane of incidence, the amount reflected would be sin2 (ir), while if polarized in a plane perpendicular to it A= sin2 (i+r)'
the proportion would be B=
tan2 (i-r) tan2 (1+r)'
i and r representing
the angles of incidence and refraction respectively.
light may be regarded as composed of two equal beams polarized at right angles, hence the amount reflected
R = ↓ (A + B) = ↓ (sin2 (i + r)
a formula which may be applied to any special case by substituting proper values for i and r. The value of A evidently increases as i varies from 0° to 90°. That of B, on the other hand, diminishes from 0° until i+r=90°, when it equals 0; or at this angle, which is that of total polarization, all of the ray B is transmitted, all the reflected beam being polarized in the plane of incidence. When i=90°, A=1, B=1; hence all the light is
reflected. When i=0°, A, B, and R equal (~—1)2;
; hence the reflected light increases with n, being zero when n=1, and 100 per cent. when n=co. Many familiar phenomena are thus readily accounted for-for instance, the brightness of the diamond, the covering power of white lead as a paint, and the brilliancy of wet or varnished stones and woods.
A curious case presents itself when n=1+dn, or differs from unity only by an infinitesimal amount. A then becomes equal to dn2 (1+tan2 i)2, and B to dn2 (1-tan2 i)2. When i=0, A, B, and R equal dn2; and this quantity is accordingly taken as the unit in Table I. The first column gives various values of the angle of incidence, the second and third the corresponding values of A and B, the fourth the amount of light reflected, and the fifth the degree of polarization. The other columns will be explained hereafter. This Table is evidently applicable to all cases where the media bounding the surface have nearly the same index of refraction, whether its absolute amount is great or small.
* Communicated by the Author.
Light reflected when ʼn is near unity, or equals 1+dn.
The most important application of Fresnel's formula is to the case of glass, where n somewhat exceeds 1.5. The first portion of Table II. gives in an abbreviated form the result of a computation for various values of i, A, B, R, and the polarization of the reflected and refracted rays. When, as frequently happens in the case of plates of glass, the ray passes through several parallel surfaces, a portion of the light reflected back by the second surface is again intercepted by the first surface. It may readily be proved that, if A is the amount reflected by a single surface, the amount transmitted including this internal reflection will be 1-A while if no internal reflection took place it would 1+ (m−1)A' be only (1-A)". In Table II. the values of A, B, R and of the polarization are given for 2, 8, and 20 surfaces, corresponding to 1, 4, and 10 plates of glass. In all these cases the index n=1.55.