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parallel to the plane A B ; then the force b c being parallel to the plane, has no action on it; c a, therefore, represents the full force acting on the plane. This force being again resolved into the two forces, a d, in the direction of gravity, and cd, in the direction of the fluid; c d, will then represent the portion of the original force ba, which acts on A B, in the direction of its motion, and the force a d, has no other tendency than to raise the body or plane out of the fluid, or act in a direction contrary to gravity. If, then, the plane or body in motion forms an angle, inclined to the direction of the fluid, in proportion as the velocity of the fluid, or the force on the plane increases, the tendency of the body to rise will increase also. By the common rules of plane trigonometry, any of these forces, or the whole combined, can be easily expressed in analytic forms, and the rules of the Calculus as easily applied to them when necessary.
This upward force, which appears to be entirely overlooked by writers on this subject, and which is to the direct force acting on the plane as the cosine of the angle of inclination to radius, accounts for the boats with which Mr. Coraham and Mr. Fairbairn made their experiments, skimming on the surface of the water, at high velocities, their gravity being so much counteracted, and, therefore, scarcely forming a ripple. Hence, whatever be the law of the forces, this upward force will be constantly varying the portion of surface exposed to the immediate actiou of the fluid, and, therefore, constantly diminishing the surface, and of course the resistance as the velocity increases, when the plane inclines as A B, in the figure, to the direction of the fluid, F A.
Should the keel, or the whole body of the vessel be raised at a small angle, this upward force would then come into play ou the whole of that portion of the vessel immersed in the fluid, which is inclined to it. Hence, important questions arise, not only as regards the shape of the prow, but likewise the whole of the vessels, and whether those calculated for high velocity ought not to be constructed, or loaded, so as to sink the stern a small portion ; for with such velocities the effect would be considerable. Thus, if we suppose the inclination of the plane, A B equal 45°, half the entire force, or i bard a, will act in a direction contrary to gravity, or in raising the body. The upward pressure is greatest at this angle, and equal at equal elevations above and below it ; so that at 30° and 60° the upward pressures are equal ; as also at 1° and 89o. These conclusions can be easily derived from the figure, or by calculation. If, for further elucidation, we suppose the plane to pass through the fluid with a force that would cause it to pass over a space of thirty-two and one-sixth feet in a second of time, and to be inclined to the direction of the fluid at an angle of 45°, then the upper pressure would be equal to a force that would cause the body to pass over sixteen and a half feet in a second, and, therefore, an exact counterpoise to the force of gravity. In this case, the plune or body would have no pressure on the fluid. A greater force than this, or a force that would cause the plane, in this position, to pass over more than twenty-two miles an hour, would cause it to mount above the fluid, until the action of gravity preponderating, would bring it back to its own element again. At an angle of 30°, or of 60°, it would require a velocity of little more than thirty-seven feet in a second, or about twenty-five miles an hour, to counteract the force of gravity; and at an angle of 1°, or of 89°, the velocity required, for counteracting gravity, would be nine hundred and iwelve and a half feet per second, or six hundred and twenty-eight miles an hour. *
The examples here selected, are similar to what we observe in a kite held by a string, and mounting in the air ; its plane forming an angle with the direction of the wind ; and it is on the same principle, that birds, as buzzards, &c. can float for a considerable time in the air, without moving their wings, after having previously acquired a considerable velocity ; owing to the peculiar shape of their breast. When there is no velocity forward, as in the case of a hawk eyeing a bird, the wings must be agitated with considerable quickness to support its weight against the force of gravity.
* These conclusions are very different from those deduced by Dr. Lardner and Mr. Fairbairn, from theory. In the examples exhibited above, in place of the resistance vanishing altogether as we have pointed out; they would make it increase with the squares of the velocities; so that in the last example, it would become, according to them, equal to (628)2 in miles; or equal to a force that would counteract a velocity of 394,384 miles an hour.
It is well known that it would require more than a velocity of 25,993.3 feet in a second, or 17,7224 miles an hour, for a locomotive engine on a level rail-road, to detach itself by its velocity from the road, abstracting from the effects of the air's resistance, admitting that it met with no impediment, and the radius of the earth 21,000,000 feet, and the force of gravity, such as to cause a beavy body to descend 16,087 feet in a second. (See Cavallo's Philosophy, American edition, vol. 1, p. 73, or Wallace on the Globes and Practical Astronomy, p. 473. art. 46.) The locomotive, while describing this immense space of 25,993.3 feet in a second, its friction from its weight would only be such as would cause it to deviate from its tangent, or direction in a straight line, 16 1'2 feet, during this time. And this would be the measure of its friction from gravity, in a second, whatever be the space described in that time; were it only one mile, or even one foot. This remarkable fact, which depends on the principle, that a body once put into motion, will retain that motion for ever, until counteracted by an equal and contrary motion, explains almost the whole doctrine of the friction of locomotive engines, as far as gravity is concerned. The application of this simple principle would save correspondents in the London Mechanics Magazine (N. A. series) a vast deal of labour in their unsuccessful investigations. It is in consequence of this law that the bird, or baloon, in the air, is not abandoned by the earth notwithstanding its rapid and various motions in space ; and that a variety of other phenomena are satisfactorily explained.
If, on the contrary, we suppose the fluid to act on the plane, A B, and g a to represent the force with which a film of it acts on the portion a, in that direction, the same construction being made as in the preceding case, f a will represent the force on the plane, and e a the force acting downwards, or in the direction of gravity. An example of the action of this force may be found in mill-dams. When they form an angle of 45° with the current, the pressure downwards is equal to half the full force of the water striking them, and this action increases with the velocity of the current. This circumstance not being noticed in the French experiments, may account for some anomalies in them, owing to the shape of the prow of the vessels.
It must now, we think, be acknowledged, that Dr. Lardner did not reason philosophically, and that Mr. Fairbairn did not consult theory, in their vieus of this subject.
But what we have mentioned, is far from giving the elements even of a complete theory. For, in the most simple case of all, that of a plane ; besides the inclination we have been speaking of, it may have another in a direction oblique to the current, or direction of the motion, as in the case of the rudder of a vessel ; and each side of the prow is in a similar situation. Here, again, the same allowances are to be made, and the same circumstances will take place as regards this second inclination. In euch case there will be a force acting upwards.
When we consider the variety of these circumstances, and many more which have not been touched upon, the difference in the actions, when they doexist, between those of incompressible and compressible or elastic fluids (in the theory of which much error exists as yet,j the variety of planes, of curve surfaces, and of solids that may present themselves, and consider, at the same time, that as yet, the most simple case of the problem has not been satisfactorily solved; we must be astonished at the temerity, or rather ignorance of those, who generally undertake its solution, and give their indigested mass to the world as theory.*
In the well written article on resistance, by Professor Robison, he has scarcely added any thing to our knowledge on this subject, in a scientific point of view ; but he arrives, after all his labour, which must be considerable, at this remarkable conclusion, that very few who undertake the solution of this problem, since the days of Newton, understood it, or perceived its difficulty.
* We must except from the number of those, such authors as Bossut, Bonguer and Euler, particularly the latter. Iu his “Théorie Complette de la Construction et de la Maneuvre des Vaissaux," a good translation of which into English, is given by Henry Watson, Esq, once chief engineer of Bengal; he has adopted the common theory of the square of the sines, &c. although attempted to be established somewbat differently trom the usual mode, (see pp. 74-5-6, &c. of the translation, new edition, London, 1790,) which renders the whole of the work, considered one of the most important on the subject, of little use comparatively.
The physical condition of the problem, its constant variation with the variation of various causes, every moment present new circumstances which are generally overlooked. When these circumstances, and the innumerable number and variety of the figures are considered, together with their various positions with regard to the direction of the fluid, it must, we think, be acknowledged, that this problem is by far the most difficult, which the physico-inathematical sciences have yet encountered, and that, in the investigation of which, the least progress has been made. The reason of this appears evident, for if those who attempt its solution, start out with a false hypothesis, as for example, in assuming the squares of the sines for the sines themselves-whatever dexterity they might possess in the management of the Calculus, in exhibiting general expressions, &c. the whole must ultimately end where it commenced, without establishing a single legitimate principle. · Hence, even in Newton's determination of the resistance on hemispheres, cylinders, &c. the figure of the most advantageous vessel for sailing, and various other inquiries which his followers have instituted, the physical conditions of the problems not being well understood, or attended to, the results, however elaborate, are not to be depended upon.
Mr. Thredgold, a civil-engineer, of considerable theoretical acquirements, notwitstanding the formidable aspect of this subject, bas vigorously grappled with it in the Philosophical Magazine and Annals of Philosophy, (uew series, April, 1828, pp. 249–262.) He calls his investigations, “A new Theory of the resistance of Fluids, compared with the best experiments." He has, in what he conceives a new theory, a considerable display of algebraical formulæ, with, now and then, a trace, as analytical chemists would say, of fluxions ; expended, however, to little purpose, on one or two antiquated problems, (pp. 252-3.) In all these investigations, the principle of the square of the sine of the inclination, to express the direct force on the plane, or which is the same physical principle, the cube of this sine to express the resistance in the direction of the motion, which we have shewn to be erroneous, is made use of. Besides, there is no notice taken of the pressure in the direction of, or in a direction contrary to gravity, as we have pointed out ; which, with the variation of the velocity of the body of fluid, must constantly vary, and, therefore, vary as constantly the physical state of the inquiry. His formulas, therefore, though evidently elaborate, are of no theoretical or practical utility. This is the more to be regretted, as Mr. Thredgold possesses more than an ordinary share of science in his profession, and has laboured hard to establish his new theory.
" It will be evident to any one who examines the preceding paper, (he remarks, at the conclusion of his article) that it must have cost me a great deal of labour, and, in consequence, I was desirous of presenting it to the Royal Society. But finding that I must sacrifice all claim to new theoretical investigation, in order to secure its appearance in the transactions of that society, I chose in preference to send it in to Newcastle, and to take this most respectable channel for presenting it to the public, knowing that it will be extensively circulated among men of science, as well as that, in these days it does not require the aid of authority to support the cause of truth; while recollecting the state of hydrodynamical science as it appears in books written for the use of University students, we know that when authority has not truth to propagate, it does not hesitate to teach that which is known to be erroneous.”—“Having opened a new path in this difficult subject of the motion of fluids, it was not in my nature to stand still." &c.
Here Mr. Thredgold mentions those inquiries in which he has been since engaged, and which he promises to present to the world, if his liealth permits. We fear, however, that without perceiving it, he has got himself already within the lion's grasp, and although, in this awkward situation, it may not be in his nature to stand still, yet it may cost him much greater efforts to extricate himself, than in getting into this unpleasant difficulty. We are of opinion, however, that a few lessons from his friends of the University, on the Calculus, in addition to bis stock of fluxions, such for example, as on partial differentials, definite integrals, the Calculus of variations, &c. might place him more nearly on a footing with those who have so vigorously grappled with this difficulty; at all events it would enable him to exhibit himself in a more modern and fashionable dress.
In our hasty review of this intricate subject, it was natural enough we should turn our attention to some of the voluminous productions of the older institutions of our own States, in the number of which, the University of Cambrige and the College at Yale, stand pre-eminent. We must confess, however, that we have been utterly disappointed. In the Cambridge Mechanics, by Professor Farrar, we have consulted the portion of it alloted to the “Resistance of fluids to bodies moying in them," (pp. 338-403) and find that almost the whole of it. is taken word for word from Ch. 5, vol. i. p. 537, &c. of 0. Gregory's Mechanics.