'To compute the effects of water wheels exactly, it is neceffary to know, in the first place, what is the real velocity of the water which impinges on the wheel; 2. The quantity of water expended in a given time; and, 3. How much of the power is loft by the friction of the machinery. 1. With regard to the velocity of the water, Mr Smeaton determined by experiments, with the machinery defcribed in the volume referred to, that with a head of water 15 inches in height, the velocity of the wheel is 8'96 feet in a minute. The area of the head being 1058 inches, this, multiplied by the weight of a cubic inch of water, equal to 579 of an ounce avoirdupois, gives 6126 ounces for the weight of as much water as is contained in the head, upon one inch in depth; and by further calculations derived from the machitery made ufe of, he computes, that 264'7 pounds of water defcend in a minute through the space of 1 inches. The power of the water, therefore, to produce mechanical effects in this cafe will be 3647 X 15, or 3970. From the refult of the experiment, however, it appeared that a vaft quantity of the power was loft; the effect being only to rafe 9'375 pounds to the height of 135 inches; to that the power was to the effect as 3970 to 9375X135=1266, or as 10 to 3*18. This, according to our author, muft be confidered as the greatest fingle effect of water upon underfhot wheel, where the water defcends from an height of 15 inches; but as the force of the current is not by any means exhaufted, we muft confider the true proportion betwixt the power and effect to be that betwixt the quantity of water already mentioned and the fum of all the effects producible from it. This remainder of power, it is plain, must be equal to that of the velocity of the wheel itself, multiplied into the weight of the water. In the prefent experiment, the circumference of the wheel moved with the velocity of 3°123 feet in the fecond, which answers to a bead of 1.82 inches; and this height being multiplied by 264'7, the quantity of water expended in a minute, gives 481 for the power of the water after it has paffed the wheel, and hence the true proportion betwixt the power and the effect will be as 3849 to 1266, or as 11 to 4. Thefe calculations are founded upon the known maxim in HYDROSTATICS, that the velocity of pouting water is nearly the fame with that which an heavy body would acquire by falling from an height equal to that of the refervoir, and is proved by the rifing of jets nearly to the height of their refervoirs. As the wheel revolved 86 times in a minute, the velocity of the water must be equal to 86 circumferences of the wheel; which, according to the dimenfions of the apparatus used by Mr Smea ton, was as 86 to 30, or as 20 to 7. The greatest load with which the wheel could move was 9 lb. 6 oz.; and by 12 lb. it was entirely ftopped. Whence our author concludes, that the impulse of the water is more than double of what it ought to be according to theory; but this he accounts for by obferving, that in his experiment the wheel was placed not in an open river, where the natural current, after it has communicated its impulfe to the float, has room on all fides to escape, as the theory fuppofes, but in a conduit, to which the float being adapted, the water cannot otherwis escape than by moving along with the wheel. It is obfervable, that a wheel working in this man ner, as foon as the water meets the float, recciving a fudden chéck, it rises up against the float like a wave against a fixed object, infomuch that when the fheet of water is not a quarter of an inch thick before the float, yet the sheet will act upon the whole farface of a float whose height is three inches; and confequently, was the float no higher than the thicknefs of the theet of water, as the theory alfo fuppofes, a great part of the force would have been loft by the water dashing over the float. Mr Smeaton next proceeds to give tables of the velocities of wheels with different heights of water; and from the whole deduces the following conclufions. 1. The virtual or effective head be ing the fame, the effect will be nearly as the quantity of water expended. 2. The expense of water being the fame, the effect will be nearly as the height of the virtual or effective head. 3. The quantity of water expended being the fame, the effect is nearly as the fquare of the velocity. 4. The aperture being the fame, the effect will be nearly as the cube of the velocity of the water. Hence, if water paffes out of an aperture in the fame fection, but with different velocities, the expenfe will be proportional to the velocity; and therefore, if the expenfe be not proportional to the velocity, the fection of the water is not the fame. 5. The virtual head, or that from which we are to calculate the power, bears no proportion to the head of water; but when the aperture is larger, or the velocity of the water lefs, they approach nearer to a coincidence; and confequently, in the large openings of mills and fluices, where great quantities of water are difcharged from moderate heads, the head of water, and the virtual head determined from the vclocity, will nearly agree, which is alfo confirmed by experience. 6. The moft general proportion betwixt the power and effect is that of 10 to 3; the extremes, 10 to 3'2, and 10 to 28. But it is obfervable, that where the power is greateft, the fecond term of the ratio is greatest alfo; whence we may allow the proportion fubfifting in great works to be as 3 to 1. 7. The propor tion of velocity between the water and wheel is in general about 5 to 2. 8. There is no certain ratio between the load that the wheel will carry at its maximum, and what will totally stop it; though the proportions are contained within the limits of 20 to 19, and 20 to 15 ; but as the effect approaches nearest to the ratio of 20 to 15, or of 4 to 3 whe the power is greateft either by increase of velocity or quantity of water, this feems to be the mon applicable to large works; but as the load that a wheel ought to have, in order to work to the beli advantage, can be affigned by knowing the effec that it ought to produce, and the velocity it ought to have in producing it, the exact knowledge of the greateft load it will bear is of the leaft contequence in practice. Mr Smeaton, after having finished his experiments on the underfhot mills, reduced the nun.ber of floats, which were originally 24, to 12 U 2 which which caufed à diminution in the effect, by reafon that a greater quantity of water efcaped between the floats and the floor than before; but on adapting to it a circular fweep of fuch a length, that one float entered into the curve before the other left it, the effect came so near that of the former, as not to give any hopes of advancing it by increafing the number of floats beyond 24 in this particular wheel. He next proceeds to examine the power of water when acting by its own gravity in turning an overhot wheel: "In reafoning without experiment (fays he), one might be led to imagine, that however different the mode of application is, yet that, whenever the fame quantity of water defcends through the fame perpendicular space, the natural effective power would be equal, fuppofing the machinery free from friction, equally calculated to receive the full effect of the power, and to make the moft of it: for, if we fuppofe the height of a column of water to be 30 inches, and refting upon a bafe or aperture of one inch fquare, every cubic inch of water that departs therefrom, will acquire the fame velocity or momentum from the uniform preffure of 30 cubic inches above it, that one cubic inch let fall from the top will acquire in falling down to the level of the aperture: one would therefore suppose that a cubic inch of water, let fall through a space of 30 inches, and there impinging upon another body, would be capable of producing an equal effect by collifion, as if the fame cubic inch had descended through the fame space with a flower motion, and produced its effects gradually. But however conclufive this reafoning may feem, it will appear, in the courfe of the following deductions, that the effect of the gravity of defcending bodies is very different from the effect of the ftroke of fuch as are nonelaftic, though generated by an equal mechanical power. Having made fuch alterations in his machinery as were neceffary for overfhot wheels, our author next gives a table of experiments with the apparatus fo altered. In these the head was 6 inches, and the height of the wheel 24 inches; fo that the whole defcent was 30 inches; the quantity of wa. ter expended in a minute was 963 pounds; which, multiplied by 30 inches, gives the power 2900: and after making the proper calculations, the effect was computed at 1914; whence the ratio of the power to it comes to be nearly 3 to 2. If, however, we compute the power from the height of the wheel only, the power will be to the effect nearly as 5 to 4. From another fet of experiments the following conclufions were deduced: 1. The effective power of the water must be reckoned upon the whole defcent: because it must be raised to that height in order to be able to produce the fame effect a fecond time. The ratios between the powers so estimated and the effects at a maximum, differ nearly from 4 to 3, and from 4 to 2. Where the heads of water and quantities of it expended are the leaft, the proportion is nearly from 4 to 3; but where the heads and quantities are greatest, it comes nearer to that of 4 to 2; fo that by a medium of the whole the ratio is nearly as 3 to 2. Hence it ap pears that the effect of overfhot wheels is nearly double to that of undershot ones; the confe quence of which is, that non-claftic bodies, when acting by their impulfe or collifion, communicate only a part of their original impulse, the remainder being spent in changing their figure in confe quence of the ftroke. The ultimate conclufion is, that the effects as well as the powers are as the quantities of water and perpendicular heights multiplied together respectively. 2. By increafing the head, it does not appear that the effects are at all augmented in proportion; for by raising it from 3 to 11 inches, the effect was augmented by lefs than one feventh of the increafe of perpendicular height. Hence it follows, that the higher the wheel is in proportion to the whole defcent, the greater will be the effect; becaufe it depends lefs upon the impulse of the head, and more upon the gravity of the water in buckets: and if we confider how obliquely the water iffuing from the head muft ftrike the buc kets, we fhall not be at a lofs to account for the little advantage that arifes from the impulfe there of, and shall immediately fee of how little confcquence this is to the effect of an overfhot whee This, however, as well as other things, must be fubject to limitation; for it is neceffary that the velocity of the water fhould be fomewhat greater than the wheel, otherwife the latter will not only be retarded by the ftriking of the buckets againft the water, but fome of the power will be loft by the dashing of the water over the buckets. 3. To determine the velocity which the circumference of the wheel ought to have in order to produce the greateft effect, Mr Smeaton obferves, that the more flowly any body descends by the force of gravity when acting upon any piece of machinery, the more of that force will be spent upon it, and confequently the effect will be the greater. If a ftream of water falls into the buc ket of an overfhot wheel, it will be there retain ed till the wheel discharges it by moving round and of confequence, the flower the wheel moves the more water it will receive; fo that what is loft in velocity is gained by the greater preffure of water upon the buckets. From the experi ments, however, it appears, that when the whee made about 20 turns in a minute, the effect was greateft; when it made only 18, the motion was irregular; and when loaded fo as not to admit its turning 18 times, the wheel was overpowered with the load. When it made 30 turns, the power was diminished by about one twentieto and when the number of turns was increased to 40, it was diminished by one fourth. Hence w fee, that in practice the velocity of the whee should not be diminished farther than what wil procure fome folid advantage in point of power becaufe, ceteris paribus, the buckets must be lar ger as the motion is flower; and the wheel being more loaded with water, the ftrefs will be pro portionably increafed upon every part of the work The beft velocity for practice therefore will be that when the wheel made 30 turns in a minute which is little more than three feet in a fecond This velocity is applicable to the higheft overthe wheels as well as the loweft. Experience how ever determines, that high wheels may deviat further further from this rule before they will lofe their power, by a given aliquot part of the whole, than low ones can be permitted to do; for a wheel of 24 feet high may move at the rate of 6 feet per fecond; while our author has feen one of 33 feet high move very steadily and well with a velocity of little more than two feet. The reafon of this fuperior velocity in the 24 feet wheel, may probably be owing to the small proportion that the head requifite to give the proper velocity to the wheel bears to the whole height. 4. The maximum load for an overfhot wheel is that which reduces the circumference of the wheel to its proper velocity; which is known by dividing the effect it ought to produce in a given time by the space intended to be described by the circumference of the wheel in the fame time: the quotient will be the refiftance overcome at the drcumference of the wheel, and is equal to the load required, including the friction and refiftance of the machinery. 5. The greatest velocity that an overshot wheel is capable of, depends jointly upon the diameter or height of the wheel and the velocity of falling bodies; for it is plain that the velocity of the circumference can never be greater than to defcribe a femi-circumference, while a body let fall from the top defcribes the diameter, nor even quite fo great as the difference in point of time muft always be in favour of that which falls through the diameter. Thus, fuppofing the diameter of the wheel to be 16 feet and an inch in diameter, an heavy body would fall through this space in one fecond; but fuch a wheel could never arrive at this velocity, or make one turn in two feconds, por could an overfhot wheel ever come near it; because, after it has acquired a certain velocity, great part of the water is prevented from entering the buckets, and part is thrown out again by the centrifugal force: and as these circumstances have a confiderable dependence upon the form of the buckets, it is impoffible to lay down any general rale for the velocity of this kind of wheels. 6. Though in theory we may fuppofe a wheel to be made capable of overcoming any refiftance whatever, yet as in practice it is neceffary to make the wheel and buckets of fome certain and determinate fize, we always find that the wheel will be ftopped by fuch a weight as is equal to the effort of the water in all the buckets of a femicircumference put together. This may be determined from the ftructure of the buckets themelves; but in practice, an overfhot wheel becomes unferviceable long before this time; for when it meets with fuch an obftacle as diminishes its velocity to a certain degree, its motion becomes irregular; but this never happens till the velocity of the circumference is lefs than the two feet per fecond, when the refiftance is equable. 7. From the above obfervations, we may eafily deduce the force of water upon breaft-wheels, &c. But in general, all kinds of wheels where the water cannot defcend through a given space unlefs the wheel moves with it, are to be confidered as overhot wheels; and thofe which receive the impulfe or fhock of the water, whether in an hori. zontal, oblique, or perpendicular direction, are to be confidered as undershots. Hence a wheel in which the water ftrikes at a certain point below the furface of the head, and after that defcends in the arch of a circle, preffing by its gravity upon the wheel, the effect of fuch a wheel will be equal to that of an underfhot whofe head is equal to the difference of level between the furface of the water in the refervoir and the point where it ftrikes the wheel, added to that of an overshot, whofe height is equal to the difference of level between the point where it ftrikes the wheel and the level of the tail-water. In the 66th volume of the Tranfactions, our author confiders fome of the caufes which have produced disagreements and difputes among mathematicians upon this fubject. He obferves, that foon after Sir Ifaac Newton had given his definition," that the quantity of motion is the meafure of the fame, arifing from the velocity and quantity of matter conjointly," it was controverted by his cotemporary philofophers. They maintained, that the measure of the quantity of motion fhould be estimated by taking the quantity of matter and the fquare of the velocity conjointly. On this fubject he remarks, that from equal impelling powers acting for equal intervals of time, equal augmentations of velocity are acquired by given bodies when they are not refifted by a medium. Thus a body defcending one fecond by the force of gravity, pafles through a space of 16 feet and an inch; but at the end of that time it has acquired a velocity of 32 feet 2 inches in a fecond; at the end of two feconds, it has acquired one that would carry it through 64 feet 4 inches in a fecond. If, therefore, in confequence of this equal increase of velocity, we define this to be a double quantity of motion generated in a given time in a certain quantity of matter, we come near to Sir Ifaac's definition; but in trying experiments upon the effects of bodies, it appears, that when a body is put in motion, by whatever caufe, the impreffion it will make upon an uniformly refifting medium, or upon uniformly yielding fubftances, will be as the mafs of matter of the moving body multiplied by the fquare of its velocity. The queftion therefore properly is, whether those terms, the quantity of motion, the momenta, or forces of bodies in motion, are to be efteemed equal, double, or triple, when they have been generated by an equable impulse acting for an equal, double, or triple time? or that it fhould be measured by the effects being equal, double, or triple, in overcoming refiftances before a body in motion can be stopped? For, according to the meaning we put upon thefe words, the momenta of equal bodies will be as the velocities or fquares of the velocities of the moving bodies. Though by a proper attention to the terms employed, however, we will find both these doctrines to be true; it is certain that fome of the most celebrated writers upon mechanics have fallen into errors by neglecting to attend to the meaning of the terms they ufe. DESAGULIERS, for inftance, after having been at pains to show that the difpute, which in his time had fubfifted for 50 years, was a difpute merely about words, tells us, that both opinions may be eafily reconciled in the following cafe, viz. that the wheel of an different diforders, who by being obliged to travel SECT. III. Of MILLS. MILL, in the proper fenfe of the word, fignifies achine for grinding corn, though in a more ral fenfe it is applied to all machines which have an horizontal circulatory motion. Mills are Linguifhed by particular names, fometimes ates from the powers by which they are moved, fometimes from the ufes to which they are ped. Hence they are called hand-mills, horfe, water-mills, fulling-mills, wind-mills, cornlevigating-mills, boring-mills, &c. The mot fimple of these is the band-mill, repriented Plate CCXII. fig. 90, where A and B represent the two stones between which the corn round, and of which the upper one, A, turns ad, but the lower one, B, remains fixed and oveable. The upper ftone is five inches Lack, and 21 inches broad; the lower one fomet broader. C is a cog-wheel, having 16 or 18 rs, which go into the trundle F, having 9 fpokes ed to the axis G, the latter being firmly inferted o the upper ftone A, by means of a piece of a. His the hopper into which the corn is put; the hoe to carry it by little and little through a at K, in betwixt the ftones, where being round into meal, it comes out through the eye L. Both ftones are inclofed in a circular wood. afe, of fuch a fize as will admit the upper one run freely within it. The under furface of the per tone is cut into grooves, as reprefented at which enable it to throw the meal out at the L more perfectly than could be done if it was e plain. Neither of them are entirely flat, upper one being somewhat concave, and the one convex. They nearly touch at the edges, but are at fome diftance in the middle, in order to let the corn go in between them. The under stone is fupported by ftrong beams, not reprefented in the figure; the fpindle G ftands on the beam MN, which lies upon the bearer O. One end of this bearer refts upon a fixed beam, and the other has a ftring fixed to it, and going round the pin P, by the turning of which the timbers O and MN may be raised or lowered, and thus the ftones put nearer, or removed farther from each other, in order to grind fine or coarse. When the corn is to be ground, it must be put into the hopper by little at a time. A man turns the handle D, and thus the cog-wheel and trundle are carried round alfo together with the ftone A. The axis G is angular at K; and as it goes round, fhakes the fhoe I, and makes the corn fall gradually through the hole K. The upper ftone going round grinds it, throwing out the meal, as already faid, at the eye L. Another handle, if thought proper, may be put at the other end of the handle E. The fpindle muft go through both ftones, in order to reach the beam MN, and the hole through which it paffes is faftened with leather or wood, fo that no meal can pass through. Mr Emerson, from whom this account is taken, obferves, that "it is a pity fome fuch mills are not made at a cheap rate, for the fake of the poor, who are much diftreffed by the roguery of the millers." The conftruction of a horse-mill differs not from that of the hand-mill juft defcribed, excepting that inftead of the handle D, the fpindle is furnished with a long horizontal lever and cogged wheel, which turns the trundle and ftones, as already mentioned.-The ftones are much heavier than in the hand-mill. The mills moft commonly in ufe for grinding corn are water-mills, the conftruction of which is not effentially different from that of the hand or horfe-mills. The lower mill-ftone, as already mentioned, is fixed, but the upper one moveable upon a fpindle. The oppofite furfaces of the two ftones are not flat, but the one convex and the other concave, though in a very small degree. The upper ftone, which is fix feet in diameter, is hollowed only about an inch in the middle, and the other rises of an inch. They approach much nearer each other at the circumference, and the corn begins to be ground about two thirds of the radius diftant from the circumference, and there it makes the greatest resistance, the space between the two ftones being in that place only about or of the thickness of a grain of corn; but as thefe ftones, as well as thofe of the hand-mill or horfe-mill, can be feparated a little from each other, the meat may be made fine or coarse in them, as well as in the two former mills. To cut and grind the corn, both the upper and under ftones have furrows cut in them, as is obferved in the hand-mill. Thefe are cut perpendicularly on one fide, and obliquely upon the other, by which means each furrow has a fharp edge, and by the turning of the ftones, the furrows meet like a pair of fciffars, and by cutting the corn, make it grind the more eafily. They are cut the fame way in both ftones when they lie upon their backs, by which means they run cross |