XLII. On a Salt that is invisible in its Mother-liquor. MANY years ago Sir David Brewster devised a simple but accurate method for determining the refractive power of solid fragments without the trouble of grinding and polishing them. For this purpose a broken chip of the solid, so irregular that no object could be seen through it, was put into a fluid of the same refractive power, in which case the incident rays would suffer no refraction in passing from the fluid into the solid or from the solid into the fluid; consequently objects could be seen distinctly through the broken irregular chip. Thus a bit of crown glass, of very irregular shape, so as to appear almost opaque, became nearly invisible when put into Canada balsam, and so transparent that a printed page could be easily read through it. By mixing fluids of different refractive powers it is not difficult to obtain a compound of the same refractive density as that of the solid we wish to test. Oil of cassia mixed with oil of olives in different proportions may be used for determining the refractive powers of all solids from 5 077 (that of oil of cassia) to 3.113 (that of oil of olives). I am not aware whether this valuable suggestion has ever been adopted by persons who deal in or work up precious stones. If a rough topaz or other rough stone be put into Canada balsam, oil of sassafras, or other fluid of nearly the same refractive density, and be turned round so that the rays of light may pass through in every direction, the slightest flaws or cracks are readily detected. Even when the refractive density of the stone exceeds that of any fluid, as in the case of diamond, jasper, spinelle, ruby, and some others, yet by immersing them in oil of cassia or terchloride of antimony flaws and imperfections not visible and not suspected start into view. Even when examined in water, flaws are more perceptible than when seen in air. By this method also precious stones may be distinguished from pastes. I do not remember any case recorded by chemists in which a salt has the same refractive density as the liquor in which it is formed, and is consequently invisible in it. Such a case occurred to me last winter while examining the action of low temperatures on supersaturated solutions, chiefly of double salts. The zinc and sodic sulphates were mixed in atomic proportions, dissolved in a very small quantity of water, only just enough to prevent the anhydrous salt from being thrown down during the boiling; the boiling solution was filtered into clean test-tubes, and was protected from the action of nuclei by plugging the tubes with * Communicated by the Author, having been read before the Chemical Section of the British Association at Liverpool, September 20, 1870. cotton-wool. When cold the tubes were put into a freezing-mixture at 10° F., and afterwards into one at 0° F., apparently without any effect. The tubes were set aside with the cotton-wool undisturbed, and they remained at rest during a week. On again examining them the cotton-wool was removed; but there was no sign of crystallization until, on closing one of the tubes with the thumb and inverting it, a large mass of crystals became visible in consequence of the draining off of the mother-liquor, now only a saturated solution. Air entered some cavities in the crystals, and on turning back the tube so as to allow the mother-liquor once more to envelope the crystals, these air-filled cavities stood out with most perfect definition, while the crystals themselves again became invisible. This experiment very much impressed me with the value of Sir David Brewster's suggestions; and I cannot fancy a better test in the hands of an intelligent lapidary for detecting flaws and cavities in precious stones before deciding on their value or proceeding to cut and polish them. On repeating the experiment with the double salt, I found that at about 0° F. the solution became solid, but so transparent that no casual observer would suspect it to be so. One of the tubes that was more than three-fourths full had a few scattered needles at the surface, showing that crystallization had set in. On passing down a platinum spatula the solution was found to be pulpy, so much so that the tube could be inverted without any loss of liquid. By repose the pulpy salt became crystalline, and the mother-liquor, of the same refractive density, separated. This sodio-zincic sulphate, as obtained in an open evaporatingdish, contains only four equivalents of water. In a closed tube, if left to repose during some weeks, it assumes a different state of hydration; and as it does so it acquires a different index of refraction as compared with that of the mother-liquor, and so becomes visible. Highgate, N., September 8, 1870. XLIII. On the Geodesic Lines on an Oblate Spheroid. THE HE theory of the geodesic lines on an oblate spheroid of any excentricity whatever was investigated by Legendret ; and the general course of them is well known, viz. each geodesic line undulates between two parallels equidistant from the equator (being thus either a closed curve, or a curve of indefinite length, * Communicated by the Author. † Mém. de l'Inst. 1806; see also the Exer. de Calcul Integral, vol. i.(1811) p. 178, and the Traité des Fonctions Elliptiques, vol. i. (1825), p. 360. Phil. Mag. S. 4. Vol. 40. No. 268. Nov. 1870. according to the distance between the two parallels); at a point of contact with the parallel the curve is, of course, at right angles to the meridian; say this is V, a vertex of the geodesic line, and let the meridian through V meet the equator in A; the geodesic line proceeds from V to meet the equator in a point N, the node, where AN is at most =90°; and the undulations are obtained by the repetition of this portion VN of the geodesic line alternately on each side of the equator and of the meridian. N Fig. 1. α I consider in the present paper the series of geodesic lines which cut at right angles a given meridian A C, or, say, a series of geodesic normals. It may be remarked that as V passes from the position A on the equator to the pole C, the angular distance  N increases from a certain determinate value (equal, as will appear, to 90°, if C, A are the polar and equatorial axes respectively) up to the value 90°; and it thus appears that, attending only to their course after they first meet the equator, the geodesic normals have an envelope resembling in its general appearance the evolute of an ellipse (see fig. 1 and also fig. 2), the centre hereof being the point B at the distance BA=90°, and the axes coinciding in direction with the equator B A and meridian BC: this is in fact a real geo desic evolute of the meridian CA. The point a is, it is clear, the intersection of the equator by the geodesic line for which V is consecutive to the point A (so that ZBOA= = (1-2)90°); ); and the point y is the in tersection of the meridian C B by the geodesic line for which V is consecutive to the point C; and its position will be in this way presently determined. I was anxious, with a view to the construction of a drawing and a model, to obtain some numerical results in relation to a spheroid of considerable excentricity, and C I selected that for which (polar axis=≥ equatorial), Before proceeding further, I remark that Legendre's expression "reduced latitude" is used in what is not, I think, the ordinary sense; and I propose to substitute the term "parametric latitude:" viz., in figure 3, referring the point P on the ellipse by means of the ordinate MP Q to a point Q on the circle, radius OK (=OA, fig. 1), and drawing the normal PT, then we have for the point P the three latitudes, Fig. 3. x=4QOK, parametric latitude; T M K viz. N' is the parameter most convenient for the expression of the values of the coordinates x, y (x=A cos N', y=C sin λ') of a point P on the ellipse. The relations between the three latitudes are The course so that X", X', λ are in the order of increasing magnitude. I use in like manner l, l', l" in regard to the vertex V. of a geodesic line is determined by the equation cos sin a=const., where X is the reduced latitude of any point P on the geodesic line, and a is at this point the azimuth of the geodesic line, or its inclination to the meridian. Hence, if l' be the parametric latitude of the vertex V, the equation is cos λ' sin a= cos l' (whence also, when λ'=0, a=90°—l'; that is, the geodesic line cuts the equator at an angle =l', the parametric latitude of the vertex). The equation in question, cos λ'sin a=cos l', leads at once to Legendre's other equations: viz. taking, as above, A, C for the equatorial and polar semiaxes respectively, and 8 for the C2 excentricity, 8=√1-C; and to determine the position of P on the meridian, using (instead of the parametric latitude X') the angle determined by the equation and writing, moreover, s to denote the geodesic distance VP, and A to denote the longitude of P measured from the meridian CA which passes through the vertex V, these are which differential expressions are to be integrated from 4=0; and the equations then determine X', s, and A, all in terms of the angle 4, that is, virtually s and A, the length and longitude, in terms of the parametric latitude X'. Hence integrating from 40, and using the notations F, E, II of elliptic functions, we have s= C M E(c, $), A=={(n+c2) II(n, c, 4) — c2F(c, $)}; viz. these belong to any point P whatever on the geodesic line, parametric latitude of vertex =l'; and if we write herein &=90°, then they will refer to the node N, or point of intersection with the equator. |