Mr. Smeaton considers the above to be the maximum single effect of water upon an undershot wheel, where the fall is fifteen inches. The remainder of power, it is plain, must be equal to that of the velocity ofthe wheel itself, multi plied into the weight of the water, which, in this case, brings the true proportion between the power and the effect to be as 3.849 to 1.166 ; or as 11 to 4.
Where a wheel revolved 86 times in a mi nute, the velocity of the water must have been equal to 86 circumferences of the wheel; which, according to the dimen sions 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 would move was 915. 6o:; by 1215. it was entire ly stopt. From this we are to conclude, as Mr. Smeaton did, that the impulse of the water is more than double what our theory states it to be. This he accounts for by the wheel being placed in a narrow slit ; so that the water could not escape but by passing with the wheel's motion ; thus giving a multiplied force. Further, it is to be remarked, that when a float board comes in contact with the water, it receives a certain check, which causes the back or upper part of the float-board to become loaded with a kind of wane, which accumulates in consequence of the momentary impediment, and consequent ly adds to the impetus. This added force must ever be in proportion to the depth to which the float-board sinks into the stream; not exceeding its whole depth beyond the rim, or body, of the wheel to which it is attached.
The following conclusions result from the velocities of wheels, as acted op. on by different heights of water. 1. The head, or altitude, being the same, the ef fect will be proportioned to the quantity of water expended ; or, in other words, according to the weight and velocity of the impinging fluid. The expence or quantity of water being the same, the ef fect will be .nearly in proportion to the height of the head. 3. The quantity of water expended being the same, the ef fect is nearly as the square of the veloci ty. 4. The aperture whence the fluid issues being the same, the effect will be nearly as the cube of the velocity. Hence, if water passes out of an aper ture in the same section, but with differ ent velocities, the expence will be pro portioned to the velocity ; therefore, if the expence be not proportioned to the velocity, the section of the water cannot be the same. 5. The virtual bead, or that from which we calculate the power, bears no proportion to the bead-water; but when the aperture is larger, or the velocity of the water less, they ap proach nearer to a coincidence; come quently, in the large openings of mills and sluices, where great quantities of water are discharged from moderate areas, the head of water, and the virtual head (de termined from the velocity) will nearly agree, as experience proves. 6. The most
general proportion between the power and the effect was 10 to 3 ; the extremes are It) to 3, 2, down to 10, 2.8. 7. The proportion of velocity between the water and the wheel is usually as 5 to 2. 8. Al though we have no certain maximum of the power of a wheel, that is, what it will carry, and no more, we may generally con sider the limits to be, that wheels which work freely with 15 will stop, when 20 are opposed to their motion: consequent ly, when 3 is the effect, 4 will stop the work. But in general we find it extreme ly difficult to ascertain this point ; though in works that are perfectly well executed, and where the powers, with the resist ances, are judiciously computed, the quantity of the latter necessary to pro duce equilibrium, which here amounts to cessation, will be found to correspond with that scale.
Speaking of float-boards, it may be proper to state, that they should be ra ther numerous than few. Mr. Smeaton found, that in undershot mills, when he reduced the number of floats from twen ty-four to twelve, the effect was reduced one half, because the water escaped be tween the floats without touching them ; but when he added a circular sweep of such length, that before one float-board quitted it another had entered it, he found the former effect nearly restored.
This mode more particularly applies to breast-wheels, or such as receive the wa ter immediately below the level of the ax is. In such the circular trough is indispen sable ; because the water would not com municate the full effect desirable from the joint operations of velocity and weight. In this kind of wheels it is proper that the float-boards should be confined, both at their sides and at their extremities, so that the water may accompany all the way from the head down to the lowest part of the wheel, whence it should draw off with sufficient readiness to allow the suc ceeding fall to supply its place, without being in the least retarded. It should be understood, that any quantity of water remaining in the trough, at the bottom of a breast-wheel in particular, must tend more or less to oppose its motion, in the exact ratio with the disposition of the fluid to become stagnate or stationary.