short time, and yet we should have at command a force capable of little velocity but of great effort, a machine must be found to supply the velocity necessary for the intensity of the force: if, on the contrary, the mechanist has only a weak power at his disposition, but capable of a great velocity, a machine must be adopted that will compensate, by the velocity the agent can communicate to it, for the force wanted: lastly, if the agent is capable neither of great effort, nor of great velocity, a convenient machine may still enable him to accomplish the effect desired, and make the product Fvt of force, velocity, and time, as great as is requisite. Thus, to give another example: Suppose that a man, exerting his strength immediately on a mass of 25 lbs, can raise it vertically with a velocity of 4 feet per second; the same man acting on a mass of 1000lbs, cannot give it any vertical motion though he exerts his utmost strength, unless he has recourse to some machine. Now he is capable of producing an effect equal to 25 x 4 x t: the letter t being introduced because, if the labour is continued, the value of t will not be indefinite, but comprised within assignable limits. Thus we have 25 × 4 × 1 = 1000 × v x t; and consequently v = of a foot. This man may therefore with a machine, as a lever, or axis in peritrochio, cause a mass of 1000 lbs to rise of a foot, in the same time that he could raise 25 lbs 4 feet without a machine; or he may raise the greater weight as far as the less, by employing 40 times as much time. From what has been said on the extent of the effects which may be attained by machines, it will be seen that, so long as a moving force exercises a determinate effort, with a velocity also determinate, or so long as the product of these is constant, the effect of the machine will remain the same: thus, under this point of view, supposing the preponderance of the effort of the moving power, and abstracting from inertia and friction of materials, the convenience of application, &c, all machines are equally perfect. But, from what has been shown, (props. 9, 10) a moving force may, by diminishing its velocity, augment its effort, and reciprocally. There is therefore a certain effort of the moving force, such that its product by the velocity which comports to that effort, is the greatest possible. Admitting the truth of the law assumed in the propositions just referred to, we have, when the effect is a maximum, v=w, or F = ; and these two values obtaining together, their product w expresses the value of the greatest effect with respect to the unit of time. In practice it will always be adviseable to approach as nearly to these values as circumstances will admit; for it cannot be expected expected that they can always be exactly attained. But a small variation will not be of much consequence: for, by a well-known property of those quantities which admit of a proper maximum and minimum, a value assumed at a moderate distance from either of these extremes will produce no sensible change in the effect. If the relation of F to v followed any other law than that which we have assumed, we should find from the expression of that law values of F, v, &c, different from the preceding. The general method however would be nearly the same. With respect to practice, the grand object in all cases should be to procure a uniform motion, because it is that from which (cæteris paribus) the greatest effect always results. Every irregularity in the motion wastes some of the impelling power; and it is the greatest only of the varying velocities which is equal to that which the machine would acquire if it moved uniformly throughout: for, while the motion accelerates, the impelling force is greater than what balances the resistance at that time opposed to it, and the velocity is less than what the machine would acquire if moving uniformly; and when the machine attains its greatest velocity, it attains it because the power is not then acting against the whole resistance. In both these situations therefore, the performance of the machine is less than if the power and resistance were exactly balanced; in which case it would move uniformly (art. 1). Besides this, when the motion of a machine, and particularly a very ponderous one, is irregular, there are continual repetitions of strains and jolts which soon derange and ultimately destroy the whole structure. Every attention should therefore be paid to the removal of all causes of irregularity. CHAPTER XII. PRESSURE OF EARTH AND FLUIDS AGAINST WALLS AND FORTIFICATIONS, THEORY OF MAGAZINes, &c. PROBLEM 1. To determine the Pressure of Earth against Walls. WHEN new-made earth, such as is used in forming ramparts, &c, is not supported by a wall as a facing, or by counterforts and land-ties, &c, but left to the action of its weight and the weather; the particles loosen and separate from each other, other, and form a sloping surface, nearly regular; which plane surface is called the natural slope of the earth; and is supposed to have always the same inclination or deviation from the perpendicular, in the same kind of soil. In common earth or mould, being a mixture of all sorts thrown together, the natural slope is commonly at about half a right angle, or 45 degrees; but clay and stiff loam stands a greater angle above the horizon, while sand and light mould will only stand at a much less angle. The engineer or builder must therefore adapt his calculations accordingly. Now, we have already given, at prop. 44 Statics in vol. 2, 6th edition, the general theory and determination of the force with which the triangle of earth (which would slip down if not supported) presses against the wall on the most unexceptional principles, CB acting perpendicularly against AE at K, m; D or 3 AE3. AB2 OBE2 where m denotes the specific gravity of the earth of the triangle ABE.-It may be remarked that this was deduced from using the area only of the profile, or transverse triangular section ABE, instead of the prisimatic solid of any given length, having that triangle for its base. And the same thing is done in determining the power of the wall to support the earth, viz, using only its profile or transverse section in the same plane or direction as the triangle ABE. This it is evident will produce the same result as the solids themselves, since, being both of the same given length, these have the same ratio as their transverse sections. In addition to this determination, we may here further observe, that this pressure ought to be diminished in proportion to the cohesion of the matter in sliding down the inclined plane BE. Now it has been found by experiments, that a body requires about one-third of its weight to move it along a plane surface. The above expression must therefore be reduced in the ratio of 3 to 2; by which means it becomes m for the true practical efficacious pressure of the earth against the wall. AE3. AB2 9BB2 AB BE Since which occurs in this expression of the force of the earth, is equal to the sine of the AEB to the radius I, put the sine of that Ee; also put a AE the altitude of the triangle; then the above expression of the force, viz, VOL. III. AE. AB2 S AE3. AB2 9BE2 m, becomes falem, for the perpendicular pressure of the earth against the wall. And if that angle be 45°, as is usually the case in common earth, then is e2, and the pressure becomes a3m. PROBLEM II. To determine the Thickness of Wall to support the Earth. CB AG ENF In the first place suppose the section of the wall to be a rectangle, or equally thick at top and bottom, and of the same height as the rampart of earth, like AEFG in the annexed figure. Conceive the weight w, proportional to the area GE, to be appended to the base directly below the centre of gravity of the figure. Now the pressure of the earth determined in the first problem, being in a direction parallel to AG, to cause the wall to overset and turn back about the point F, the effort of the wall to oppose that effect, will be the weight w drawn into FN the length of the lever by which it acts, that is w x FN, or AEFG XFN in general, whatever be the figure of the wall. But now in case of the rectangular figure, the area GE=AF XEFax, putting a=AE the altitude as before, and r = EF the required thickness; also in this case FN EF = r, the centre of gravity being in the middle of the rectangle. Hence then axxxar, or rather arn is the effort of the wall to prevent its being overturned, n denoting the specific gravity of the wall. Now to make this effort a due balance to the pressure of the earth, we put the two opposing forces equal, that is ar2n = a3e2m, or a2n = ža2e2m, an equation which gives for the requisite thickness of the wall, just 2m to sustain it in equilibrio. Coról. 1. The factor de, in this expression, is the line AQ drawn perp. to the slope of earth BE: theref. the breadth r becomes 2m 40, which conseq. is directly propor n tional to the perp. AQ.-When the angle at E is = 45°, or half a right angle, as is commonly the case, its sine e is m a. Further, when and the breadth of the wall r = }α√. 22 the wall is of brick, its specific gravity is nearly the same as the the earth, or mn, and then its thickness ra, or onethird of its height.-But when the wall is of stone, of the specific gravity 2, that of earth being nearly 2, that is, m = 2, and n = 21; then /== '895, of which 13 m n is 298, and the breadth = 298aa nearly. That is, the thickness of the stone wall must be of its height. PROBLEM III.. To determine the Thickness of the Wall at the Bottom, when its Section is a Triangle, or coming to an Edge at Top. In this case, the area of the wall AEF CB is only half of what it was before, or only AEX EF = ax, and the weight waxn. But now, the centre of gravity is at only of FE from the line AE, or FN=FE3r. Consequently FN X w = 3xx axnax'n. This, as before, being put the pressure of the earth, gives the equation ar1n={a3e2m, or x2n={a2e2m, and the root, or thickness EF = aeNT = a√√ T the slope of 45°. H m 3n m for Now when the wall is of brick, or m = n nearly, this becomes rα=408a3a, or of the height nearly. But when the wall is of stone, or m to n as 2 to 24, then m m 6n √, and the thickness r or a =α√73= 365a a nearly, or nearly of the height. PROBLEM IV. To determine the Thickness of the Wall at the Top, when the Face is not Perpendicular, but Inclined as the Front of a Fortification Wall usually is. Here GF represents the outer face of a fort, AEFG the profile of the wall, having AG the thickness at top, and EF that at the bottom. Draw GH perp. to EF; and conceive the two weights w, w, to be suspended from the centres of gravity of the rectangle AH and the tri ENHME angle GHF, and to be proportional to their areas respectively. Then the two momenta of the weights w, w, acting by the levers FN, FM, must be made equal to the pressure of the earth in the direction perp. to AE. $ 2 Now |