theless, feel themselves compelled to admit the trap-rocks above mentioned as volcanic, from their analogy with those of the Giant's Causeway! Even such as, with Dr MacCulloch, include granite among the products of Fire, or, even with my ingenious friend Dr Boué, broadly assert that all rocks, at least as far down as sandstone, which are not of mechanical origin, must be referred to this agent, cannot but be puzzled at the resemblance subsisting between certain varieties of granite and of the old red sandstone.

Since, therefore, it seems to follow, that Fire and Water, although such opposite agents, have, in some instances, produced effects nearly, if not altogether identical, I do not see that the geologist who returns from Auvergne, persuaded that great part at least of what has been called the Newest Flatz-Trap Formation of Werner is of volcanic origin, ought to be accused of inconsistency, if he still hesitates as to the real origin of those rocks, which, if, in their external characters, they approach to the latter, would seem, nevertheless, from their repeated alternations with sandstone, and other strata of a similar description, to be of Neptunian origin themselves *.

I had intended adding some remarks on the Fresh Water Formation of Auvergne; but as the present memoir has already

Those of your readers who may recollect what I said in my former letter respecting the Puy Marmont, where trap-rocks which were assumed to be of igneous origin are described as alternating with a fresh-water limestone, may object to the present conclusion as inconsistent with the former statement. I have nowhere, however, denied the possibility of an alternation of volcanic rocks with the products of aqueous deposition, the actual occurrence of which is indeed sufficiently established, by the observations of those who have travelled in countries admitted to be volcanic; but, on the other hand, it will, I think, be allowed that the probability of such an accident diminishes in the direct ratio of the number of such alternations, and, therefore, that those who do not go so far as to consider the sienite and greenstone rocks of Primitive Districts as volcanic, and consequently see nothing in the structure and composition of hornblende or augite rocks in general, which stumps them as the Products of Fire, will regard such a succession of beds as that which meets our eye on the Fife coast, from Kinghorn to Kirkcaldy, as more probably referable to one agent, and that water, than to the alternate influence of two opposite forces, repeatedly giving way, as if by consent, one to the

other.

swelled to such a size as I fear must have exhausted the patience of many of your readers, must subscribe myself,

MAGD. COLL. OXFORD,

Jan. 16. 1821.

Very truly yours, C. DAUBENY.

}

ART. XI.-Observations on the Resistance of Fluids. By WILLIAM WATTS, Esq. Communicated by the Author.

THE effect of the deep immersion of bodies in water, still remains a contested point in the theory of the resistance of Fluids; and many persons are of opinion, that the resistance increases with the depth, although the experiments of Sir Isaac Newton, made with small globes formed of wax, having lead inclosed in them, and made so light as to weigh only a few grains in water, in order that they might descend very slowly in this fluid, seem to prove that the resistance is equal in every part; for these globes having been let fall in water, descended in the same manner as they would have descended in a fluid in which the resistance was every where equal, though, when they were near the bottom of the vessel, the compression was many times greater than when they were near the top.

This supposed increase of resistance at greater depths, is even assumed as a principle by Mr Gordon, in his Theory of Naval Architecture; but the grounds of this assumption do not appear to be satisfactorily explained; and this is not to be wondered at, if we consider that a competent knowledge of the mutual actions of fluids on each other, is accessible only to those who are intimately acquainted with all the refinements of the modern analysis, and that, without this key, there is no admittance into this department of the physico-mathematical sciences.

These observations, however, do not apply to the investigation of DON GEORGE JUAN D'ULLOA, who, in his "Examine Maritimo," has attempted to consider the subject in a scientific manner, and has given some important experiments, similar to

those adduced by M. Bouger, in his "Manœuvre des Vaisseaux," but leading to very different conclusions.

According to this author, the resistance is in the sub-duplicate ratio of the depth of immersion below the surface of the water, and the simple ratio of the velocity of the resisted surface, jointly; and my object in drawing up this paper, is to attempt to prove, that there is nothing in this proposition inconsistent with the generally received principle in experimental philosophy, that the resistances are, very nearly, in the duplicate ratio of the velocities; and, at the same time, to answer some objections which have been urged against it by the writer of the article "Resistance" of Fluids in the Encyclopædia Britannica.

In making this attempt, I have adopted the principles employed in the investigation of this problem by d'Ulloa, or rather by M. Prony, in his "Architecture Hydraulique," section 868. &c.; but in order to render the investigation more clear and obvious to readers in general, I have taken the liberty to deviate from his manner of treating it, because I am most decidedly unfriendly to the trite form of modern solutions; and as I consider the investigation, as given by M. Prony, to be defective, inasmuch as it appears, to me at least, to be left in an unfinished state, I have attempted to complete it, in the best manner I am able.

Let o be an elementary orifice, or portion of the surface of the side of a vessel filled with water; call the area of this small surface b, and let h be its depth below the horizontal surface of the fluid. Let p be the actual pressure exerted on the surface b, the density of the fluid, and g the accelerating force of gravity the velocity acquired by a heavy body, during the first second of its fall; then, by the principles of Hydrostatics, the pressure on the orifice o, when the fluid escapes into a vacuum, will be

હૃ

p = gębh + gębh.

This value of P consists of the pressure gębh', which the horizontal surface of the fluid sustains; and also of the pressure ggbh, which represents the weight of a volume of the fluid equal to bh. If, therefore, we neglect the first part gebh, it is evident, that the pressure which the surface b sustains at the depth h below the horizontal surface of the fluid, is equal to the weight

of a prism of the fluid, whose base is equal to b, and whose height is equal to h: thus, when the fluid escapes into air, the pressure at the orifice o, will be equal to gębh only; because the pressure gebh', which is transmitted to the surface b, by the intervention of the different strata of the fluid, is balanced by an equal and opposite pressure of the atmosphere acting without the orifice o. In this case, therefore, we have

p = gebh......(1).

The same reasoning holds good, when we suppose that the small surface o is immersed to the depth h below the upper surface of a stagnant fluid, and moved through it with the velocity v; for when the velocity is very great, so that a perfect vacuum is left behind the small surface o, we shall have

p = gçbh + gębh';

but when the fluid does not escape into a perfect vacuum, or any thing like it, but into a mixture of air and water, the pres sure at the depth h below the upper surface will be, nearly,

p = gebh, as before.

I have been more particular on this head than I should otherwise have been, with a view to meet an objection that has been advanced against this part of the subject.

Now, it is well known, that when the pressure is the same at the upper surface of the fluid and at the orifice o, or at the anterior and posterior surfaces of the base b, the water will flow out with the velocity u√2gh; whence we deduce u2 = 2 gh,

u2

2g

If we substitute this value of h in the equation (1), we shall have

Let us now suppose the elementary surface o to move with the velocity ; then the fluid would meet it either with the velocity u + v, or u — v, according as it moves in the direction opposite to that of the effluent fluid; or in the same direction with it: and, by substituting (u±v)2 instead of u in the preceding equation, we shall obtain

nerally considered to be the most accurate of any yet given. This great diversity in the values which different authors have deduced from their experiments for the absolute resistance of water, is very remarkable; and it should induce philosophers to exert their utmost efforts in endeavouring to detect any fallacy that may have crept into the principles or reasonings by which the result of the theory has been deduced; and in multiplying experiments, with a view to obviate the great disparity that still exists in the absolute value of the resistance, as determined by different authors. I am well aware that this cannot be accomplished, without incurring a considerable expence; and it is this consideration alone, that deters me from making the attempt, and not the difficulty of the undertaking; for, however arduous it may appear, I imagine it might be overcome by a steady perseverance and attention.

It may not be irrelevant to remark here, that when water escapes from a vessel through a small orifice, perforated in one of its sides, the effective discharge is only about 0,62 of the theoretical, owing to the contraction of the fluid vein,—a circumstance which has not been taken into the account in the preceding investigation this reduction being made in the value of R, when the mass M is assumed equal to unity, the resistance will be found NEARLY equal to the mass of a prism of water, whose base is equal to b, the area of the fluid vein, and whose height is equal to twice the fall producing the effluent velocity.

It should also be remarked, that the result of the preceding investigation is not rigorously exact, because a portion of the fluid is thrown back on the sides of the plane surface, in consequence of which, it is neither so much urged, nor so continually impelled as before; but notwithstanding this, it leads to an approximation, at which, however, we are obliged to stop, on account of the insuperable difficulties of the subject. This remark is due to Francœur. Nevertheless, this approximation is the limit to which the real phenomena of the impulse and resistance of fluids continually approach. When, therefore, the law by which the phenomena deviate from the theory, shall be once dedetermined, by a well chosen series of experiments, this ap proximate theory will become nearly as valuable as a true one; for the rules and practice of computation are established even