Note on Molecular Contraction in Natural Sulphides.

By E. J. CHAPMAN, PH. D., Professor in University College, Toronto.

(Read May 26, 1882).

1. In mixtures of metallic or mineral bodies, the composition, it is well known, may be calculated from the specific gravity of the mixture, or the specific gravity from the composition. In actual combinations, on the other hand, neither of these results, as a rule, can be deduced-a molecular contraction or expansion of the combined bodies usually, if not invariably, ensuing.

2. In the case of certain natural sulphides, very striking and apparently anomalous differences are thus manifested. In cubical Iron Pyrites, for example, the average density or specific gravity equals 5.0, whilst the percentage composition is represented by S. 53.3, Fe. 46.7. In Copper Pyrites, a more or less closely related mineral, occurring constantly under the same geological conditions, the percentage composition equals S. 34.9, Cu. 34.6, Fe. 30.5. This latter mineral, therefore (with much less of the light body, sulphur, and with heavier metallic base), should possess a priori the higher specific gravity: whereas its maximum density does not exceed 4.2 or 4.3. It is evident, consequently, that in Iron Pyrites a much greater contraction has ensued, by which more matter has been brought into a given space; or, that, in Copper Pyrites a greater molecular expansion has followed.

3. If the theoretical specific gravity of these bodies be calculated from their composition, as mixtures, it will be seen that in each case contraction has really ensued—the actual density being greater in both cases, but much greater in the case of Iron Pyrites than in Copper Pyrites. In the former (calling the actual or average density 5.06) the excess equals 2, as shewn in the table under §6 below; whilst in the latter (putting the actual density at 4.2) the excess is only 0.24. A cubic foot of Iron Pyrites contains, therefore, 124.64 lbs. more matter than it would contain if its components were merely in admixture; whilst in a cubic foot of Copper Pyrites the excess is only equal to 15.58 lbs. The excess in Iron Pyrites is, of course, equivalent to the weight of two equal volumes of water; and in Copper Pyrites to practically one-fourth of an equal volume of water.

4. In these deductions, the specific gravity of sulphur has been taken at 2.0; that of iron at 7.8; and that of copper at 8.9. A cubic foot of water has been assumed to average 62.32 lbs. The formula used in calculating the theoretical specific gravity is the well known equation :

In this equation, b and b' equal the respective densities of the bodies B., B'; and V., V'. equal, of course, the volumes of the latter.

5. Of all natural sulphides, cubical Iron Pyrites appears to present the greatest contraction. Marcasite (Prismatic Pyrites) approaches it very closely-the excess of weight in the latter species being equal to that of 13 vol. of water. In Pyrrhotine (Magnetic Pyrites) it equals one volume. In Galena, Argentite, Copper Glance, Stibnite, Bournonite, and many other sulphides quite irrespective, apparently, of atomic constitution or crystallization-it equals one-half the weight of an equal volume of water; whilst in Zinc Blende it amounts to only one-fifth, and in the arsenide Smaltine to one-eighth of that weight. In Realgar and in Greenockite (Cd S) there is apparently no contraction; nor is any revealed in the lead and silver tellurides, Altaite and Hessite.

6. As these somewhat curious relations do not appear to have been referred to in mineralogical publications, I have ventured in this brief notice to bring them before the attention of the Society. The annexed table exhibits the relations as presented by the more commonly occurring sulphides and allied compounds; but, in some cases, owing to the difficulty of obtaining absolutely accurate densities, the results are necessarily approximative only. This, however, does not in any way invalidate the general fact that, among minerals of related composition and constitution, very striking differences of molecular condensation occur. The question naturally arises as to whether these differences result merely from accidental causes, or are the outcome of some definite law. Accidental they can scarcely be― as the amount of contraction is essentially the same in examples of the same substance occurring under widely different conditions, and in widely separated localities. They would thus appear to depend upon some general law; but the nature of this law, in the present state of our knowledge, is seemingly without explanation.

TABLE SHOWING RATIO OF MOLECULAR CONTRACTION IN CERTAIN NATURAL SULPHIDES AND RELATED BODIES OF COMMON OCCURRENCE.

Many oxidized bodies appear to present far more marked examples of condensation than any recorded in the preceding Table of Sulphides. Thus, whilst the sp. gr. of Silicon does not exceed 2:49 or 2.5, the sp. gr. of Quartz equals 26 to 27. In Aluminium, again, the sp. gr. equals 26; and in Corundum (with only 523 Al) it exceeds 3.9. In these and other similar cases, therefore, either great contraction is revealed, or the metal or other base, when liberated from oxygen, must undergo enormous expansion.

Symmetrical Investigation of the Curvature of Surfaces, etc.

By ALEXANDER JOHNSON, M.A., LL.D., Dublin; Professor of Mathematics and Natural Philosophy, McGill University, Montreal.

(Read May 25 1882.)

The object of the following paper is to show that the leading propositions concerning the curvature of surfaces, may be obtained in a direct and simple manner by a symmetrical investigation, in which each proposition leads naturally to the following.

The first step to this, is a simplified solution of the well known problem, "To find the equation, referred to its axes, of the plane section of a central quadric." The paper may be considered as consisting of two parts: the first, referring to the axes of conics and quadrics; and the second, to the curvature of surfaces specially.

The following is a summary of these parts:

I. AXES OF CONICS AND QUADRICS.

1o. Symmetrical investigation of the magnitudes and directions of the axes of a plane section of a central quadric.

2o. Geometrical interpretation of the analytical conditions.

3o. Symmetrical solution of four homogeneous equations which give a symmetrical determinant.

4°. Conditions that the section of the quadric be circular.

5. Application of same method:

(a) To find magnitude and direction of the axes of a quadric.

(b) To find magnitude and direction of the axes of a conic.

(c) To the discussion of the nature of the plane sections of any quadric given by the general equation.

II. CURVATURE OF SURFACES.

6o. Investigation of the radius of curvature, at a given point of any surface, of any plane section through the point.

7°. Deduction of the value of the radius of curvature of a normal section, and Meunier's Theorem.

8°. Equation, for a given point of any surface, of a quadric such that the squares of the semi-diameters of the section of it made by the tangent plane to the surface give the radii of curvature of the corresponding sections of the surface.

9°. Value of the principal radii of curvature, and Euler's formulæ.

10°. Directions of maximum and minimum curvature.

11°. Conditions for umbilics.

12o. Lines of curvature.