neither can be seen, especially if care be taken to use excess of zirconia. If, however, the amount of uranium be very small, and so much of other oxides be present as to make the bead very dark, or too opaque from deposited crystals, before it is sufficiently concentrated for the compounds with the oxides of uranium to crystallize out, it may be impossible to detect it. In order to apply the test in the case of complex minerals, a bead of borax, boric acid, and pure zirconia should be prepared, then a small quantity of the mineral added, and, after fusion and sufficient concentration, the bead made to crystallize in the manner already described. If needleshaped crystals be not deposited in the bead when very hot, and if it do not suddenly turn opaque when reheated, the result may not be satisfactory. In this manner it is easy to detect uranium in grain of such minerals as Fergusonite, tyrite, and yttrotantalite, even when they contain no more than 1 or 2 per cent. If in such cases the spectrum of the uranous compound cannot be obtained, the bead should always be flamed and reexamined, to see if that of the uranic compound is thereby developed.

In a similar manner we may make use of a little oxide of uranium to detect zirconia; but the test is far less delicate than the converse, because it is almost impossible to obtain the compound in a crystalline state, unless there be an excess of zirconia. Not more than 1 grain of uranic oxide should be employed, or the bead may be too opaque. There is no difficulty in thus detecting zirconia in zircons, or in katapleiit; but the presence of so much of other bases in minerals like eudialyte prevents our obtaining a satisfactory result. There certainly could not be a more characteristic test to confirm the results of other methods, or to identify such a small quantity of approximately pure zirconia as could not easily be distinguished in any other way.

The only other compound of uranic oxide of very abnormal character which I have so far discovered is that with cericic oxide. So much of both oxides should be fused with borax in the oxidizing flame as will yield a bead which is perfectly clear, and of pale yellow colour when rapidly cooled, but crystallizes when gently reheated. If the constituents be present in a certain proportion, it then turns from pale yellow to a deep blue, as though coloured by oxide of cobalt. In mos cases the bead is rendered nearly opaque by the number of crystals; but sometimes, though it turns deep blue, it remains transparent, owing to the compound being set free in a state similar to that of the red oxide of copper in a borax blowpipe-bead, with carbonate of soda and oxide of tin, treated in the reducing flame. The spectrum of these blue beads shows no absorption-bands, but merely a general absorption at the red end; and it is curious to find that the combination of two yellow substances gives rise to a deep blue, in much the same manner as when the yellow ferrocyanide of potassium is added to a yellow ferric salt. The production of this blue colour on the addition of a little uranic oxide might be employed with advantage to identify moderately small quantities of cerium, even when mixed with a number of other

substances; but unfortunately the presence of much oxide of lanthanum, which is so commonly associated with it, interferes, as though the ceric oxide had a stronger affinity for the oxide of lanthanum than for uranic oxide.

The most characteristic peculiarity of the compound of yttria and uranic oxide is that it will not crystallize out from a borax blowpipe-bead, and that the affinity of the uranic oxide for yttria is stronger than for zirconia. Perhaps erbia may prove to act in the same way, but I have not been able to examine that earth quite free from yttria. On adding yttria to a bead with zirconia and a little uranic oxide, and gently flaming it in the oxidizing flame, the uranic oxide combines with the yttria and rises to the surface as an orange-coloured scum, which has a great tendency to collect on the platinum wire; and if sufficient yttria be added, the crystallized borate of zirconia is left in the interior almost colourless, and so free from uranic oxide that no absorption-bands can be seen in the spectrum. We may take advantage of this circumstance to detect yttria in small quantities of compound minerals like Gadolinite and Fergusonite; and I may here say that by combining such means with the observation of the spectra of the transparent or crystalline beads, and of the form of the crystals when slowly deposited, with or without the addition of suitable reagents, we may often detect twice as many constituents in minerals as could be accomplished by the ordinary methods of blowpipe chemistry-an advantage which I am sure will be appreciated by those engaged in the study of rocks, when it is often so important to obtain satisfactory results with small quantities of material. I have also found these methods of great practical use in examining small residues in the qualitative analysis of minerals, and have thus unexpectedly discovered small quantities of comparatively rare elements.

I have tried the effect of many other substances along with zirconia and the oxides of uranium, and find that most of them have no sensible influence, unless they are present in considerable relative quantity. The most striking effect is that of oxide of tin, which causes the two absorption-bands in the yellow and yellow end of the green in the spectrum of the uranous oxide compound to be nearly equally dark, whereas without the oxide of tin that in the yellow is comparatively faint. This is another illustration of the manner in which certain substances, having no special action on light, influence by their presence the properties of another. The oxides of uranium are unusually sensitive to such actions, and thus not only lend themselves to us as blowpipe-reagents, but also seem more than any others to afford the means of explaining the relation between the physical conditions of compounds and their action on light.

The only compound of zirconia with any other oxide to which I need now draw attention is that with chromic oxide, as deposited from a borax blowpipe-bead. After treatment in the deoxidizing flame, when the cooled very pale-green bead is gently reheated, this 'compound crystallizes out * See my paper, Monthly Microscopical Journal, vol. i. p. 349.

so as to give a fine red-pink colour by transmitted light, even when so little chromium is present that the glassy bead is scarcely at all green. If too strongly heated the pink tint is lost. This compound is of interest in connexion with the colour of rubies and other minerals coloured red by chromic oxide. To others, like the emerald, it imparts a green colour, and on the whole it acts on light in such a variable manner according to the presence of other substances, that the spectra may be made use of as a means of identifying particular minerals, though they do not present anything like such striking anomalies as those met with in the compounds of zirconia with the oxides of uranium.

II. "On the Mathematical Theory of Stream-lines, especially those with four Foci and upwards." By WILLIAM JOHN MACQUORN RANKINE, C.E., LL.D., F.R.SS. Lond. and Edinb., &c. Received January 1, 1870.

(Abstract.)

A Stream-line is the line that is traced by a particle in a current of fluid. In a steady current each individual stream-line preserves its figure and position unchanged, and marks the track of a filament or continuous series of particles that follow each other. The motions in different parts of a steady current may be represented to the eye and to the mind by means of a group of stream-lines.

Stream-lines are important in connexion with naval architecture; for the curves which the particles of water describe relatively to a ship, in moving past her, are stream-lines; and if the figure of a ship is such that the particles of water glide smoothly over her skin, that figure is a streamline surface, being a surface which contains an indefinite number of stream-lines.

The author in a previous paper proposed to call such stream-lines Neoïds; that is, ship-shape lines.

He

The author refers to previous investigations relating to stream-lines, and especially to those of Mr. Stokes, in the Cambridge Transactions for 1842 and 1850, on the "Motion of a Liquid past a Solid," and of Dr. Hoppe, on the "Stream-lines generated by a Sphere," in the Quarterly Journal of Mathematics for 1856, and to his own previous papers on "Plane Waterlines in Two Dimensions," in the Philosophical Transactions for 1864, and on "Stream-lines," in the Philosophical Magazine for that year. states that all the neoïd or ship-shape stream-lines whose properties have hitherto been investigated in detail are either unifocal or bifocal; that is to say, they may be conceived to be generated by the combination of a uniform progressive motion, with another motion consisting in a divergence of the particles from a certain point or focus, followed by a convergence either towards the same point or towards a second point. Those which are

continuous closed curves when unifocal are circular, and when bifocal are blunt-ended ovals, in which the length may exceed the breadth in any given proportions. To obtain a unifocal or bifocal neoïd resembling a longitudinal line of a ship with sharp ends, it is necessary to take a part only of a stream-line, and then there is discontinuity of form and of motion at each of the two ends of that line.

The author states that the occasion of the investigation described in the present paper was the communication to him by Mr. William Froude of some results of experiments of his on the resistance of model boats, of lengths ranging from three to twelve feet. A summary of those results is printed at the end of a Report to the British Association on the "State of Existing Knowledge of the Qualities of Ships." In each case two models were compared together of equal displacement and equal length; the water-line of one was a wave-line with fine sharp ends, that of the other had blunt rounded ends, each joined to the midship body by a slightly hollow neck— a form suggested, Mr. Froude states, by the appearance of water-birds when swimming. At low velocities, the resistance of the sharp-ended boat was the smaller; at a certain velocity, bearing a definite relation to the length of the model, the resistances became equal, and at higher velocities the round-ended model had a rapidly increasing advantage over the sharpended model.

Hence it appeared to the author to be desirable to investigate the mathematical properties of stream-lines resembling the water-lines of Mr. Froude's bird-like models; and he has found that endless varieties of such forms, all closed curves free from discontinuity of form and of motion, may be obtained by using four foci instead of two. They may be called from this property quadrifocal stream-lines, or, from the idea that suggested such shapes to Mr. Froude, cycnoïds; that is, swan-like lines*.

Those lines are not to be confounded with the lines of a yacht having at a distance the appearance of a swan, which was designed and built some years ago by Mr. Peacock, for the figure of that vessel is simply oval.

The paper contains four chapters. The first three are mainly cinematical and geometrical, and relate to the forms of stream-line surfaces in two and in three dimensions, especially those with more than one pair of foci and surfaces of revolution, to the methods of constructing graphically and without calculation, by means of processes first applied to lines of magnetic force by Mr. Clerk Maxwell, the traces of such surfaces, which methods are exemplified by diagrams drawn to scale, and to the motions of the particles of liquid past those surfaces. The fourth chapter is dynamical : it treats of the momentum and of the energy of the disturbance in the liquid, caused by the progressive motion of a solid that is bounded by a ship-shape stream-line surface of any figure whatsoever; of the ratio borne by the total energy of the disturbance in the liquid to that of the disturbing body when that body displaces a mass of liquid equal to its own mass,

* Κυκνοειδής.

which ratio ranges in different cases from to 1; of the acceleration and retardation of ships as affected by the disturbance in the water, and espe→ cially of the use of experiments on the retardation of ships in finding their resistance; and of the disturbances of pressure which accompany the disturbances of motion in the liquid. Up to this point the dynamical principles arrived at in the fourth chapter are certain and exact, like the geometrical and cinematic principles in the three preceding chapters. The results obtained in the remainder of the fourth chapter are in some respects approximate and conjectural, and are to a great extent designed to suggest plans for future experiments, and rules for their reduction. These results relate to the disturbances of level which accompany the disturbances of motion when the liquid has a free upper surface, to the waves which originate in those disturbances of level, and the action of those waves in dispersing energy and so causing resistance to the motion of the vessel, to friction, or skin-resistance, and the "wake" or following current which that kind of resistance causes the disturbing solid body to drag behind it, and to the action of propelling instruments in overcoming different kinds of resistance.

The resistance caused by viscosity is not treated of, because its laws have been completely investigated by Mr. Stokes, and because for bodies of the size of ships, and moving at their ordinary velocities, that kind of resistance is inconsiderable compared with skin-resistance and wave-resistance. The resistance caused by discontinuity of figure is stated to be analogous in its effects to friction, but it is not investigated in detail, because ships ought not to be built of discontinuous (commonly called "unfair”) figures.

The author in the first place calls attention to the agreement between the position of the points at which there is no disturbance of the pressure on the surface of a sphere, as deduced from Dr. Hoppe's investigation, published in 1856 (Quarterly Journal of Mathematics), and on the surface of a short vertical cylinder with a flat bottom, as determined by the experiments of the Rev. E. L. Berthon before 1850 (Proc. Roy. Soc. vol. v. 1850; also Transactions of the Society of Engineers, 6th December, 1869). The theoretical value of the angular distance of those points from the foremost pole of the sphere is sin-141° 49'; the value deduced from experiment is 41° 30'.

The author then adds some remarks on a suggestion made by Mr. William Froude, that the wave-resistance of a ship is diminished when two series of waves originating at different points of her surface partially neutralize each other by interference; and states that, with regard to this and many other questions of the resistance of vessels, a great advancement of knowledge is to be expected from the publication in detail of the results of experiments on which Mr. Froude has long been engaged.