and if a power can raise half the weight to double the height; or double the weight to half the height, in the same time that another can, those two powers are equal.* But note, all this is to be understood in case of slow or equable motion of the body raised; for in quick, accelerated, or retarded motions, the vis inertia of the matter moved will make a variation. In comparing the effects produced by water-wheels, with the powers producing them; or, in other words, to know what part of the original power is necessarily lost in the application, we must previously know how much of the power is spent in overcoming the friction of the machinery, and the resistance of the air; also what is the real velocity of the water at the instant that it strikes the wheel; and the real quantity of water expended in a given time. From the velocity of the water, at the instant that it strikes the wheel, given; the height of head produc-tive of such velocity can be deduced, from acknowledged and experimented principles of hydrostatics; so that by multiplying the quantity, or weight of water, really expended in a given time, by the height of head so obtained; which must be considered as the height from which that weight of water had descended in that given time; we shall have a product, equal to the original power of the water; and clear of all uncertainty, that would arise from the friction of the water, in passing small apertures; and from all doubts, arising from the different measure of spouting waters, assigned by different authors. On the other hand, the sum of the weights raised by the action of this water, and of the weight required to overcome the friction and resistance of the machine, multiplied by the height to which the weight can be raised in the time given, the product will: be equal to the effect of that power; and the proportion of the two products will be the proportion of the power to the effect; so that by loading the wheel with different weights successively, we shall be able to determine at what particular load, and velocity of the wheel, the effect is a maximum.. The manner of finding the real velocity of the water, at the instant of its striking the wheel; the manner of finding the value of the friction, resistance,. &c. in any given case; and the manner of finding the real expence of water, so far as concerns the following experiments, without having recourse to theory; being matters on which the following determinations depend, it will be necessary, to explain them. To determine the Velocity of the Water striking the Wheel. It has already been mentioned, in the references to the figures, that weights are raised by a cord winding round a cylindrical part of the axis. First, then, let the wheel be put in motion by the water, but without any weights in the scale; and let the number of turns in a minute be 60; now it is evident, that were the wheel free from friction and resistance; that 60 times the circumference of the wheel would be the space through which the water would have moved in a minute; with that velocity wherewith it struck the wheel: but the wheel being incumbered by friction and resistance, and yet moving 60 turns in a minute, it is plain, that the velocity of the water must have been greater than 60 circumferences before it met with the wheel. Let now the cord be wound round the cylinder, but contrary to the usual way, and put a weight in the scale; the weight so disposed, which may be called the counter-weight, will endeavour to assist the wheel in turning the same away, as it would have been turned the water: put therefore as much weight into the scale as, without any water, will cause it to turn somewhat faster than at the rate of 60 turns in a minute: suppose 63; let it now be tried again by the water, assisted by the weight; the wheel therefore will now make more than 60 turns; suppose 64; hence we conclude the water still exerts some power in giving motion to the wheel. Let the weight be again increased, so as to make 644 turns in a minute without water: let it once more be tried with water as before; and suppose it now to make the same number of turns with water as without, viz. 64; hence it is evident, that in this case the wheel makes the same number of turns in a minute, as it would do if the wheel had no friction or resistance at all; because the weight is equivalent thereto; for were it too little, the water would accelerate the wheel beyond the weight; and if too great, retard it; so that the water now becomes a regulator of the wheel's motion; and the velocity of its circumference becomes a measure of the velocity of the water. In like manner, in seeking the greatest product, or maximum of effect; having found by trials what weight gives the greatest product, by simply multiplying the weight in the scale by the number of turns of the wheel, find what weight in the scale, when the cord is on the contrary side of the cylinder, will cause the wheel to make the same number of turns the same way, without water; it is evident that this weight will be nearly equal to all friction and resistance taken together; and consequently, that the weight in the scale, with twice* the weight of the scale, added to the back or counter-weight, will be equal to the weight that could have been raised, supposing the machine had been without friction or resistance; and which multiplied by the height to which it was raised, the product will be the greatest effect of that power. The Quantity of Water expended is found thus: The pump made use of for replenishing the head with water was so carefully made, that no water escaping back by the leathers, it delivered the same quantity of water at every stroke, whether worked quick or slow; and as the length of the stroke was limited, consequently the value of one stroke (or on account of more exactness 12 strokes) was known, by the height to which the water was *The weight of the scale makes part of the weight both ways.-Orig. thereby raised in the head; which being of a regular figure was easily measured. The sluice by which the water was drawn upon the wheel, was made to stop at certain heights by a peg; so that when the peg was in the same hole, the aperture for the effluent water was the same. Hence the quantity of water expended by any given head, and opening of the sluice, may be obtained: for by observing how many strokes a minute was sufficient to keep up the surface of the water at the given height, and multiplying the number of strokes by the value of each, the water expended by any given aperture and head in a given time will be given. These things will be further illustrated by going over the calculus of one set of experiments. Specimen of a Set of Experiments. The sluice drawn to the 1st hole. The water above the floor of the sluice.... Strokes of the pump in a minute..... The head raised by 12 strokes. The wheel raised the empty scale, and made turns in a minute 80 85 2. 5 0. 6 0. 7 0. 8 0. 361. 333. .30 . 26. 22 163. ceased working. 2361 240 maximum. 238 Counter-weight for 30 turns without water, 2 oz. in the scale. N. B. The area of the head was 105.8 square inches. Circumference of the cylinder, 9 inches. Circumference of the water-wheel, 75 ditto. Reduction of the above Set of Experiments. The circumference of the wheel 75 inches, multiplied by 86 turns, gives 6450 inches for the velocity of the water in a minute; of which will be the velocity N. B. When the wheel moves so slow as not to rid the water so fast as supplied by the sluice, the accumulated water falls back upon the aperture, and the wheel immediately ceases moving.-Orig. in a second, equal to 107.5 inches, or 8.96 feet, which is due to a head of 15 inches; and this we call the virtual or effective head. The area of the head being 105.8 inches, this multiplied by the weight of the water of the inch cubic, equal to the decimal .579 of the ounce avoirdupois, gives 61.26 ounces for the weight of as much water as is contained in the head, on 1 inch in depth, of which is 3.83 pounds; this multiplied by the depth 21 inches, gives 80.43 lb. for the value of 12 strokes; and by proportion, 39, the number made in a minute, will give 264.7 lb. the weight of water expended in a minute. Now as 264.7 lb. of water may be considered as having descended through a space of 15 inches in a minute, the product of these two numbers 3970 will express the power of the water to produce mechanical effects; which were as follows. The velocity of the wheel at the maximum, as appears above, was 30 turns a minute; which multiplied by 9 inches, the circumference of the cylinder makes 270 inches; but as the scale was hung by a pulley and double line, the weight was only raised half of this, viz. 135 inches. The weight in the scale at the maximum....... ... 8 lb. O oz. 10 0 12 9 6 or lb. 9.375. Now as 9.375 lb. is raised 135 inches, these two numbers being multiplied together, the product is 1266, which expresses the effect produced at a maximum : so that the proportion of the power to the effect is as 3970: 1266, or as 10: 3.18. But though this is the greatest single effect producible from the power mentioned, by the impulse of the water on an undershot wheel; yet as the whole power of the water is not exhausted by it, this will not be the true ratio between the power of the water, and the sum of all the effects producible from it: for as the water must necessarily leave the wheel with a velocity equal to the wheel's circumference, it is plain that some part of the power of the water must remain after quitting the wheel. The velocity of the wheel at the maximum is 30 turns a minute; and consequently its circumference moves at the rate of 3.123 feet a second, which answers to a head 1.82 inches; this being multiplied by the expence of water in a This is determined on the common maxim of hydrostatics, that the velocity of spouting waters, is equal to the velocity that a heavy body would acquire in falling from the height of the reservoir; and is proved by the rising of jets to the height of their reservoirs nearly.-Orig. minute, viz. 264.7 lb. produces 481 for the power remaining in the water after it has passed the wheel: this being therefore deducted from the original power 3970, leaves 3489, which is that part of the power which is spent in producing the effect 1266; and consequently the part of the power spent in producing the effect, is to the greatest effect producible by it, as 3489: 1266:: 10:3.62, or as 11 to 4. The velocity of the water striking the wheel has been determined to be equal to 86 circumferences of the wheel per minute, and the velocity of the wheel at the maximum to be 30; the velocity of the water will therefore be to that of the wheel, as 86 to 30, or as 10 to 3.5, or as 20 to 7. The load at the maximum has been shown to be equal to 9 lb. 6 oz. and that the wheel ceased moving with 12 lb. in the scale: to which if the weight of the scale be added, viz. 10 ounces,* the proportion will be nearly as 3 to 4 between the load at the maximum and that by which the wheel is stopped. It is somewhat remarkable, that though the velocity of the wheel, in relation to the water, turns out greater than of the velocity of the water, yet the impulse of the water in the case of a maximum is more than double of what is assigned by theory; that is, instead of of the column, it is nearly equal to the whole column. It must be remembered therefore, that in the present case, the wheel was not placed in an open river, where the natural current, after it has communicated its impulse to the float, has room on all sides to escape, as the theory supposes; but in a conduit or race to which the float being adapted, the water cannot otherwise escape than by moving along with the wheel. It is observable, that a wheel working in this manner, as soon as the water meets the float, receiving a sudden check, it rises up against the float, like a wave against a fixed object; insomuch that when the sheet of water is not a quarter of an inch thick before it meets the float, yet this sheet will act on the whole surface of a float, whose height is 3 inches; and consequently were the float no higher than the thickness of the sheet of water, as the theory also supposes, a great part of the force would have been lost, by the water's dashing over the float. * The resistance of the air in this case ceases, and the friction is not added, as 12 lb. in the scale was sufficient to stop the wheel after it had been in full motion; and therefore somewhat more than a counterbalance to the impulse of the water.-Orig. + Since the above was written, I find that Professor Euler, in the Berlin Acts for the year 1748, in a memoire intitled, Maximes pour arranger le plus avantageusement les machines destinées à elever de l'eau par le moyen de pompes, page 192. § 9. has the following passage; which seems to be the more remarkable, as I don't find he has given any demonstration of the principle therein contained, either from theory or experiment; or has made any use of it in his calculations on this subject."Cependant dans ce cas puisque l'eau est reflechie, & qu'elle decoule sur les aubes vers les cotés, elle y exerce encore une force particuliere, dont l'effet de l'impulsion sera augmenté; & experience Y Y VOL. 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