In August 1904 I discovered in the Lengeubach quarry a specimen on which were three small, round-shaped crystals, about the size of small shot, grown upon a prismatic crystal of sartorite. I measured them, and owing to their small size experienced some difficulty in determining the position of some of the faces with the goniometer at my disposal, which was of the ordinary single-circle pattern.

In the following notes I have endeavoured to give in a convenient and simple form a complete account of the relations connecting different twinning-laws or operations, which, when applied to the same crystal-structure, produce indistinguishable results.

A rigorous analysis of the possible relations between the compound structures of twin-crystals, based on the principles of the equality of the structural distances in the plane of composition, will be found in the author's papers, ' The geometry of twin-crystals' (Proc. Roy. Soc. Edinburgh, 1912-1918, vol. xxxii, pp. 416-432, 433-457) and 'Die Geometrie der Zwillingskrystalle' (Zeits. Kryst. Min., 1918, vol. lii, pp. 827-371). See also the paper 'Twin-planes and cross-planes' in this Magazine (1910, vol. xv, pp. 390-397).

There are two distinct kinds of image which are the subject of optical investigation: object-images, in which the form of the object is reproduced, and directions-images, in which every point represents a direction followed by light before reaching the objective. An instrument that shows a directions-image may be described as a hodoscope. It is focused not on the object but on infinity, so that parallel rays are collected at the same point in the image.

The Pilbara goldfield lies in the north-west division of Western Australia within the tropics, and has been described in considerable detail by A. Gibb Maitland in Bulletin 40 of the Geological Survey of Western Australia. Though the chief mineral product at present is gold, the district abounds in other economic minerals, notably ores of copper and tin, and has already become famous as being by far the most important producer of tantalum in the world, the chief source of supply of this metal being the manganotantalite of Wodgina.

In the Geological Magazine for 1918 there appeared three papers discussing the origin of tile system of cracks ill septarian nodules. Dr. A. Morley Davies opened the discussion in a paper where, after a summary of earlier literature, he again put forward an older idea. Without adducing evidence, he considered :

1. (a) that interposition of particles between those of the original matrix caused expansion which set up tensile strain in the growing body ;

2. (b) that the cracking of the nodule was due to the relief of those tensile stresses.

The material examined forms part of the late Dr. Henry S. Peele's collection, which shortly before his death he decided to present to the British Museum. Dr. Poole was at one time Government Inspector of Mines in Nova Scotia, and his collection is rich in Nova Section ores and minerals from many well-defined localities.

The geometrical representation of the faces and edges of a crystal is obtained (i) by drawing lines through a fixed point O perpendicular to each crystal-face meeting a fixed sphere with centre O in a point (face-pole) representing that face, (ii) by drawing lines through O parallel to each crystal-edge meeting the sphere in a point (edge-pole) representing that edge. This representation is inconvenient as being in three dimensions, and therefore it is customary to map the sphere on a plane. This is usually done by means of the stereographic or gnomonic projection.

Consider the poles of the faces of a crystal and the zones (great circles) which join them. Suppose we have given their gnomonic projection on to a certain tangent-plane to the fundamental sphere on which these poles and zones lie. We can find the gnomonic projection on any other plane by aid of the gnomonic net, as explained in this Magazine (1904, vol. xiv, p. 19); and similarly for the stereographic or orthographic projection.

The determination of the direction in which the light illuminating a point in a directions-image (the so-called image in convergent light) was travelling when it reached the objective has hitherto been carried out by first measuring the position of the point in the image and thence calculating the angular direction it represents. It is usually assumed that the distance of a point in the image from the centre of the field is in proportion to the sine of the angle between the path of the light that illuminates the point and the axis of the microscope.

This form of goniometer was designed in order to provide for the use of students of elementary crystallography, at as moderate a cost as possible, an efficient instrument for the measurement of the interracial angles of crystals. It is of the customary type, in which the direction of reference is given by the reflection of a distant object in a small plane mirror without optical aid, and the axis of the grHduated circle is horizontal. The mirror measures about 45 by 25 ram. and its frame is rigidly attached to the ball of a ball and socket joint, so that it is adjustable in any direction; the socket is split, and a screw linking the halves enables it to grip the ball firmly.

Suppose a crystal has two good cleavages, and that a thin section is cut in a random direction. Cleavage-cracks are seen in the section which consist of two sets of parallel lines inclined at an acute angle ϕ, which may have any value whatever, depending on the direction in which the section is cut. The question suggests itself: what is the chance that ϕ may lie between assigned limits? This question proves to be incapable of a direct mathematical solution, but it can be solved graphically. This is worked out here for two typical cases, and the general method of performing the graphical solution in every case is indicated.

Last year, while working through the geographical collection of rocks in the Geological Department of the Imperial College, I found an interesting series of volcanic rocks from the neighbourhood of the Lucalla River in the district of Cambambe, Angola, which hitherto has not been described. The specimens were collected about the year 1860 by J. J. Monteiro, a former student of the Royal School of Mines, who devoted many years to the exploration of Angola. The collection, though consisting of only ten specimens z comprises porphyritic basalts, alkalitrachytes, a nephelinite allied to monchiquite, and an augite-andesite.

The substance was met with in the course of work on steel-furnace slags, a general account of which has been communicated to the Iron and Steel Institute. These slags were for the most part melts consisting principally of the three oxides FeO, MnO, and SiO2, and yielded the silicates fayalite and rhodonite. The small amount of calcium silicate usually present is held in isomorphous solid solution ill these silicates, but when the amount of lime exceeds about eight per cent. a new silicate appears, which does not seem to belong to any of the known mineral groups.

Certain optical properties of crystals, and more particularly the refractive index, may be determined either in the directions-image, often referred to as the 'image in convergent light', or in the ordinary object-image in which the object itself is seen. In the former case, in which the index of refraction is 'usually determined by means of the critical angle of total-reflection, every point in the image corresponds to a single direction of propagation of the wave-front through the crystal-structure and to the two corresponding directions of vibration. One of these can, however, be eliminated by the insertion of a nicol in an approximate position, and thus all ambiguity in the determination of the refractive index is removed.

The pale celestite-blue calcite which occurs on the eastern half of Lot 13, Range V, of North Burgess township, Lanark county, Ontario, contains very remarkable lattice-like inclusions of a hydrous magnesium silicate.

The calcite forms crystalline masses belonging to the Grenville series, which has a very extensive areal development in certain parts of Ontario. The texture varies from medium to very coarse, in which the individual cleavage-rhombs of calcite measure one inch or more in diameter. Repeated twinning on e (0112) or (101) is almost invariably exhibited, and is especially pronounced in the more coarsely crystallized varietiea Both the twinning and the crystalline character are results of the very intense metamorphism to which this Archaean limestone has been subjected.

The mineral described in this paper occurs at tile Hudson Bay zinc mine about five miles east of Saline, near Nelson in the West Kootenay district of British Columbia, I regret that I have no data as to the exact character of the ore-body except that it occurs in a region where palaeozoic rocks are intruded by biotite-granite and that it is probably genetically connected with the intrusion of the latter. The zinc occurs ill the form of the silicate and tb a less extent the carbonate. There is also a small amount of oxidized lead-ore, principally cerussite.

The specimens of the mineral examined were obtained from a limestone quarry near the town of Renfrew. The tourmaline occurs near the contact of grey granite-gneiss and crystalline limestone of the Grenville series. The gneiss is intrusive into the limestone, and contact effects are observable both in the intrusive and in the intruded rock. The gneiss consists of quartz, plagioclase, micrccline, and a green pyroxene. The limestone is composed (almost entirely) of calcite, but contains some graphite. The secondary minerals occurring in both limestone and gneiss are, besides the tourmaline, pyrrhotite, titanite, and muscovite.