graduated circle a disc of brass having three slots, or one slot and a three-faced hole in it, in which the levelling screws of the vertical goniometer would rest. The work was left in an incomplete state, and only notes for the paper, the table of measurements and a somewhat rough stereographic projection were found. I have thrown these notes into a slightly more complete form and have made an accurate stereographic projection (see Plate) on a larger scale. In this projection the centre gives the zero of the horizontal circle, and the distance of any point from the centre gives the reading of this circle. The zero of the vertical circle is at the top of the projection, and the readings of this circle proceed from this point round to the left. Poles at a distance from the centre exceeding 90° fall below the paper and would truly be situated outside the primitive circle. By supposing the position of the eye changed to the extremity of the diameter perpendicular to the primitive above the paper, these poles will fall within the primitive; and their positions on this supposition are shown by surrounding the pole by a circlet. It will be noticed that poles 43 and 145, 55 and 140, 57 and 139, and 67 and 135, are very nearly above one another. They would be accurately superposed if the plane of projection were a plane of symmetry with respect to these poles. Prof. Miller has left no note of the way in which he actually made the observation; but it is clear that he had considerable confidence in the accuracy of the observations, and that the two goniometers were well adjusted. His usual plan was to bring the image of a bright signal (a minute triangle or line) reflected from each face of a zone into coincidence with that of a faintly illuminated vertical line as seen in a vertical blackened mirror. The readings of the two circles for any face are those given when the image of the bright signal was superposed on that portion of the faint line which lay in the horizontal plane through the bright signal. There are thirty-nine poles below the paper none of which are parallel to those above. This would be manifest at once during the observations, for such planes would be successively adjusted by rotating the vertical circle alone through 180o. Testing the zonal relations by means of the stereographic projections I have only succeeded, with such trial as I have given it, in finding about four or five poles to be common to any one zone. Were the bead a simple or twin crystal belonging to the cubic system (that accepted for platinum) it is almost certain that a much larger number of planes would lie in zones. I give such zones as I have observed by inclosing the numbers of the poles in square brackets, and append a query (?) whenever the pole does not accurately lie in, but is very near to, a zone; [94, 90, 51, 52, 141], [96, 67, 76, 75, 138, 88 (?)], [52, 30, 22 (?), 26, 71, 123], [59, 30, 43, 73 (?)], [42, 30, 33, 54 (?)], [94, 75, 78 (?), 124, 125], [140,

85, 44 (?), 39, 59 (?)]. The angles (30, 42) 27°. 44'; (30, 33) = 20°. 51′; (33, 54) = 28°. 4′3 have been calculated. Assuming them to be in a zone, the anharmonic ratio was found to =1. Other angles were determined approximately from the stereographic projection, but none of those found seemed to be near those between the common planes of cubic crystals: (71) and (69) are nearly at 90° to (30), as also the planes (119) and (136) below the paper.

The following are Professor Miller's notes which I have done no more than edit.

W. J. LEWIS.]

In 1874, I received from Major [now Colonel] Ross what appeared to be a bead of platinum, of approximate diameter 0.73 mm., which he had fused with the aid of a blowpipe, and which on cooling exhibited a large number of crystal-faces. The measurement of the crystal was undertaken under the impression that it would be comparatively easy, but the small size of the faces, which are not arranged in zones, rendered their identification all but impossible. The following method of determining their position was therefore adopted. The bead was attached to the axis of a small Wollaston's goniometer in a convenient position but without any special orientation. The small goniometer was then secured in an upright position on the graduated horizontal plate of the large goniometer so that the crystal lay in the intersection of their axes. The position of all the faces, which were not obscured by the vertical circle, could now be determined, by reference to the circles of the two goniometers, in the same way that the positions of stars can be referred to two planes by means of an altazimuth. In this way 147 faces, occupying somewhat more than a hemisphere were determined, and a stereographic projection of their poles made, in which the centre represents the zero of the horizontal circle, whilst the zero of the vertical circle is placed at the top of the projection and the angles read towards the left. As far as examined, the bead seems to differ from a crystal, (1) in that the planes do not seem to occur in pairs of parallel ones, (2) in that they are not arranged in zones. The relations of the planes to one another cannot therefore be very simple.

(2) On a crystal of Stephanite from Wheal Newton. By Prof. W. J. LEWIS.

Last summer Prof. Warrington Smyth was good enough to lend me for examination a remarkably fine crystal of stephanite, which he had obtained some years previously from Wheal Newton in Cornwall, and which is, I believe, the only specimen so far known from an English locality. It is now placed in the Jermyn Street Museum. It is implanted on a large fragment of matrix of chalybite and quartz about four inches long by about two and a half wide. The specimen was fixed for the observations in the apparatus described by Prof. Miller in a paper on his goniometer in the Phil. Magazine, 5th series, vol. II., p. 281, 1875. From the size of the matrix, and the weight of the specimen, considerable difficulty was experienced in the accurate adjustment of the zones, and no great attention was paid to centreing the edges when once the zoneaxis was placed vertical. Most of the faces gave good definite reflexions; but various artifices had to be resorted to for the purpose of obtaining some of the measurements. I was quickly struck with the smallness of the errors in the angles observed when compared with the angles given in Miller's Mineralogy. These errors are mainly due to two causes: (1) the error arising from inaccurate coincidence of the two signals, (2) that due to the eccentricity of the crystal. The first was probably, from the difficulties of the observations, at times considerable, and seemed likely to be sufficient to account for the whole error. It would also vary greatly in the measurement of different angles as the matrix would come more in the way at one time than at another. The latter I had not up to that time carefully estimated, but it seemed capable of approximate determination. I have thought the examination of this error of sufficient interest to append to this paper, and the more so as it is an error which has called forth a good deal of criticism in Germany. Schrauf in his Lehrbuch der Phys. Mineralogie, vol. I. p. 210, has given an expression for it which gives quite an erroneous idea, and seems applicable only to an utterly impossible method of observation.

The forms, and the relation of the principal zones, observed on the crystal are shown on the stereographic projection (fig. 1). The crystal is a combination of the forms

a(100), b (010), m (110), π (310), p(111), ≈(112), n(113), c(001), t (203), e (101), d (201), h (412), s (314), and λ (011).

The crystal has a peculiar hemimorphic development due to the preponderance of one set of the planes p, z, n over the others of

the same forms. This is to a certain extent shown in the diagram (fig. 2), which is a projection of the crystal on the plane c.

(fig. 2).

b

λ

m

The

crystal is, however, not so wide across in comparison to its length as is shown in this diagram. The crystal is, moreover, much traversed by twin laminæ and twin striations, which are not similarly developed on like planes, or even on all parts of one and the same plane. Thus the two planes p are finely and regularly striated, as shown in the diagram, except in a small portion of one of them adjoining the plane λ. The large plane p' has a few striations, or rather thin laminæ traversing it, perpendicular to the edge [p' m']. A well-marked lamina traverses the small planes p', z', n'. Few