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tion to the horizon, or to the direction E A, of the fluid, being equal to the angle B A E, or A B D; then the fluid acting obliquely on the plane, its direct force will be to its force acting obliquely (as demonstrated in mechanics) as radius to sine of the angle of inclination; taking r for radius, and s for sine of this angle, we have r: s:: A B, or A C,: A D. The radius being a constant quantity, or in natural numbers equal to unity, the force of the fluid acting obliquely on the plane will vary as the sine of the angle of inclination ; and this we consider to be the true expression for the direct force on the plane. But every author who has written on this subject, we believe without a single exception, which is remarkable enough, makes this force vary as the square of the sine of the inclination. For they consider that the body of fluid acting on CA, in this position is as the depth C A, admitting C A, to be entirely immersed in the fluid, and the portion acting on A, B, in the position A, B, only as the depth D A,; the portion of the water of which CD, is a section, having no action on A B. These respective portions, viz. A C, and A D, being also as r to s, and combining this proportion with the above, the direct force on both these accounts would be as r? : sạ, the square of the sine of inclination.* It must, however, appear evident that there is the same surface or portion of the fluid, or as many particles in contact with the plane A B, as with the plane A C, in their respective positions; but that those particles acting on A B, act obliquely, and therefore, the diminution of force on A B, must be from this cause alone; and hence this force must vary as the sine, and not as the square of the sine of the inclination, the latter law being evidently established on a false assumption.
If two planes, as represented by A C and A F in the figure, be equally immersed in the fluid or to the level C F according to the advocates of the squares of the sines of inclination, the
# Whatever holds with regard to the inclinations of the physical lines or sec. tions, A C,; A B, in the figure, may be easily shewn to hold with regard to any plane wbatever be its figure, or whether bounded by straight, or curve, or mixed lines. For the figures may be divided into indefinitely small sections, or elementary planes, and as the same law or proportion must hold in each of those whatever that law may be, each being acted on by an elementary filament of the stream or fluid; it will then be r (or -2) is to s (or s2) as the force on the elementary plane of the one to that on the other; or by taking the sum of the antecedents and consequents, which are the areas of the respective planes; the forces on these planes will, therefore, be in the same proportion. We may in this manner compare the force on any polygonal surface presented to the fluid to that of a section perpendi. cular to the fluid; or that on a curve surface, by resolving the curve surface into its elementary planes. Here a beautiful display of the Calculus might be made, but hitherto from pot sufficiently attending to the physical conditions of the problein, althongh a vast deal of Calculus has been exhibited instead of principles or facts, the whole has resulted only in having given some single proposition agreeing, perhaps, with some single experiment, and no other.
quantity of fluid acting on each, would be the same. But it is evident that the fluid in contact with B F is to that in contact with A C as the respective areas of those plains. But as there is so great a mass of authority in favour of the old principle, the result, as regards the sine, which in the present inquiry we consider new, must, as we hold ourselves responsible for the truth of it, be also tested by experiments, and we have consulted them.
Mr. Bland, in his Hydrostatics,* after establishing in his way, the hypothesis of the squares of the sines of inclination has the following remark in a note.
" It appears from experiments (says he) that the resistance which arises from oblique impulses do not vary as the sin.” of the angle of inclination; but that when the angles are between 50° and 90°, the common theory may be used as an approximation, observing (even in these cases) that it always gives the resistances a little less than experiment, and as much less as the angle differs from 90°.” p. 195.
Now, the tabular natural sines and cosines being decimal fractions of the radius unity, the squares of these sines, must be less than the sines themselves. The cubes and higher pow. ers must be still less; and, hence, the resistance must diminish as these powers increase. These powers, also, evidently diminish as the angles become less than 90°.
Professor Robison observes, in his very able article on resistance, in the Encyclopedia Britannica, (unquestionably one of the best articles on this subject,) that “the resistances do, by no means, vary as the sines of the angles of incidence.” And, yet, Robison establishes the common theory of the squares of the sines, on the usual hypotheses; which we have shewn to be false, and which does not agree with well conducted experiments, because it is not the correct theory. Professor Robison continues to observe, that
" As this is the most interesting circumstance, having a chief influence on all the particular modifications of the resistance of fuids, and as on this depends the whole theory of the construction and working of ships, and the action of water on our most important machines, and seems more immediately connected with the mechanism of fluids, it merits a very particular consideration. We cannot do a greater service than by rendering more generally known the excellent experiments of the French Academy.” p. 102.
After exhibiting these experiments, which were made with so much care, that the accumulation against the fore-part of
* Under the term Hydrostatics, Mr. Bland includes also, Hydraulics, Hydrodynamics and Pneumatics. This work was published in 1824, for the use of the students of the Cambridge University, England.
the vessel was carefully noticed, as well as the diminished pressure behind the vessel, his inference is as follows:
“But we see that the effects of the obliquity of incidence deviate. enormously from the theory, and that this deviation increases rapidly as the acuteness of the prow increases. In the prow of 60°, the deviation is nearly equal to the whole resistance pointed out by the theory, and in the prow of 12° it is nearly forty times greater than the theoretical resistance."
It would be only twenty-five times greater by taking the sine of 12° in place of its square. There must, therefore, be some mistake in the forty ; unless the prows were suffered to sink under the water, or that the water had accumulated over them. This circumstance would produce this extra resistance, as we shall presently shew, but it is not mentioned. After exhibiting similar deductions from Mr. Robin's and Chevalier Borda's experiments, Professor Robison further remarks
“ In short, in all the cases of oblique plane surfaces, the resistances were greater than those which are assigned by the theory. The theoretical law, (viz: the squares of the sines) agrees tolerably with observation, in large angles of incidence, that is, incidences not differing very far from the perpendicular ; but in more acute prows, the resistances are more nearly proportional to the sines of incidence, than to their squares.”
Here is abundant evidence from experiments, that these forces are proportional to the sines of inclination, and not to their squares, as we have pointed out from actual theory. Simple as this law is, when discovered, it has produced more confusion and disappointment in both elaborate investigations, and expensive experiments and undertakings than perhaps any other portion of physical science.
D'Alembert, after exhausting, we might say, the resources of the Calculus on this subject, in his “ Essai sur la resistance des Fluids," and afterwards in tom. 1, 5 and 8 of his “ OpusGules,” where he has given no less than ten extensive“ Memoires" on this subject, arrives, at length, at the following conclusion, in sec. xiii. p. 210 of his last volume.
+ The French experiments were made with fifteen boxes or vessels, each two feet wide, two feet deep, and four feet long. One was a parallelopiped, the others had prows of a wedge form, the angle varying by 12° to 180°, so that the angle of inclination (of incidence, used hy Robison in the same sense) increased by 6o from each other. These boxes were dragged across a very large bason of smooth water, in which they were immersed two feet, by means of a line passing over a wheel connected with a cylinder, from which the actuating weight was suspended. The angle of the prow or wedge was placed at the surface, so that there were two feet of the vessel immersed, and two feet above the water. (See the figures in the art.)
“ Voila (says he) ce qui résult de principes ordinare de la mécanique, appliqués à l'action des fluides sur les corps. Mais l'experience n'est pas conforme à ce resultat ; car elle prouve que l'action d'un fluide n'est pas comme le quarré des sinus des angles d'incidence."
It is astonishing that his sagacity, which managed with so much facility, and wielded with so much dexterity, the powers of the Calculus, could not perceive, at first sight, the fallacy of the reasoning in establishing this false principle; but it is still more astonishing that it should be adopted contrary to reason and experiment, in every work of science on this subject, from Newton's, even down to the present time. This appears to be a further confirmation of that strange anomaly in the human mind, that the simplest truths, and the simplest inodes of arriving at them, are the last perceived. This observation may, in some degree, diminish our surprise, that D'Alembert should conclude his last Memoire on this portion of physics. (Opus. tom. viii. p. 230,) in these mystical expressions :
“ On voit par ces details combien il est difficile de trouver une équation (x+y =1)-0(2-YV-1)=2 M v- 1, qui représent exactement les filets du fluides, au moins si l'on veut avoir une théorie rigoureuse de la resistance du fluide au movement du corps."
This famous equation of D'Alembert, arrived at, after so much labour, may, for the sake of those who are not conversant in algebraical functions, be thus translated into plain English. One imaginary quantity or difficulty, less another inaginary quantity or difficulty, equal a third imaginary quantity or difficulty.* Still, in his very last lines, he encourages geometricians to go on in further developing these imaginary functions, without bestowing a thought on the simple elementary principles on which his investigations principally depend. “ Cette matiere, (says he) paroit bein digne d'occuper les geometres.”
The celebrated Poisson, who appears to have united in himself, the analytical talents, both of Lagrange and Laplace, in the second edition of his excellent treatise on Mechanics, after making some distinctions between incompressible and elastic fluids, and giving formulas corresponding to the former, p. 481 vol. ii. and to the latter, in p. 482, arrives, at length, at the following conclusions, p. 483-4, art. 564.
" Il result de cette analyse que, soit qu'il s'agisse du movement d'un Auid incompressible, homogène ou hétérogène, ou de celui d'un fluide élastique, dont la température peut etre constante au variable suivant
* See Art. on Geometry and the Calculus, No. 1, Southern Review.
une loi donnée, on aura, dans tous les cas, un nombre d'equations égal à celui des inconues que referme le problème; mais ces equations sont, comme on voit, aux differences particules entre quartre variable indépendentas, savoir, les co-ordonnées x, y, z, et le tems t; leur integration générale est impossible par les moyens connus jusqué' ici ; et lors même qu'on parvient à les intégrer, en les simplifiant par quelqu'hypothise particuliere, il reste à déterminer, d'après l'état du fluide à l'origine du movement, les fonctions arbitraires que leurs intégrales contiennent ; ce qui presente encore de très-grandes difficultes."
We perceive,'that granting the arbitrary functions (mere supposition) in these unmanageable differentials, or which is the same, granting the elementary principles and their results; the Calculus, like Archimedes' lever, can effect any thing, notwithstanding the “tres-grandes difficultes.” But the real difficulty in the one case, consists in accurately establishing those principles, and, in the other, in actually finding the fulcrum for the lever.*
The deficiency which we have pointed out in what is called the theory or the squares of the sines, is not, after all, the principal source of error, when the planes or bodies are inclined to the direction of the fluid, and move with considerable velocities; for it is, then, principally, that the errors in the formulas, not deduced from strict theory compared with accurate experiments, exhibit themselves.
In the preceding figure, let b a represent the direction, and the full force of the fluid ; or the force resulting from the quantity of matter in motion combined with its velocity. Let this force be resolved into the two forces, a c perpendicular, and b c
* Mr J. Challis, Fellow of Trinity College, Cambridge, &c. has published in the Philadelphia Magazine tor August, 1829, pp. 123-133, some interesting remarks on the general equations given by M. Poisson in articles 567, 568 of bis Mechanics, which represent the laws of motion of incompressible fluids conducted in the most general manner; but our limits would not permit our further notice of them at present. There is here, however, no new principle developed. M. Poisson, in a very elaborate memoire, " Sur la Théorie des Andes,” published in 1818, and inserted in the Memoirs of the Royal Academy of Sciences for i826, pp. 71-186, makes the two differential equations given in vol. ii. p. 493 of bis Mechanics, the ground work of this theory of waves. In the above Memoire, there is a learned display of the Calculus, to solve a single problem, we think, after all, not so satisfactorily. There is apother, perhaps more elaborate display in a Memoire, published among those of the same academy, in 1827 (tom. vi, 1823) pp. 389-440 by M. Navier. But when we find it predicated on notions, such as the following, it does not require much investigation to shew that little reliance can be placed on it, however profound. "Nous prendrons pour principe, dans les recherches suivantes, que par l'effet du movement d'une fluide, les actions répulsives des molécules sont augmentes ou diminuées d'une quantité proportionnelle à la vitessee avec laquelle les molécules s'approchent ou s'eloignent les unes des autres." This is the fulcrum for the lever, which ought to be first established. On the same subject of waves, there are no less than 312 pages of the most refined analysis by M. A. L. Cauchy, in the “Memoires par Divers Savans," tom. 1. 1827.
VOL. VIIT. NO. 15