TABLE THIRTIETH.

Ratio of effect to power in the undershot wheel. Taken from tables S and T. (Vol. ix, Jour. Frank. Inst. p. 154-7.)

Feet.

Inches.

The average ratio of effect to power appears from this table to be .285. The ratios for the higher heads are decidedly, though not considerably, greater than for lower ones, probably on account of the greater proportion which the effect lost by the water escaping between the wheel and tail-way bears to the whole effect, and which the width of the aperture bears to the head in low heads. All those heads in which the motion of the water was affected by the mode of supplying the forebay (see Plate I. Vol. vii.) have been excluded from the table.

Taking the average ratio of the undershot wheel for comparison with that of the overshot, we find a proportion of 2.98 to one in favour of the latter wheel, approaching nearly to the ratio of three to

one.

While in the experiments of Smeaton the overshot wheel appears to have been underrated, the undershot coefficient agrees very nearly with that just deduced; hence our comparison is more advantageous. to the overshot wheel than his, in the proportion of 2.98 to 2.

The ratio of effect to power being considered as constant, it follows that in a given undershot wheel the absolute effect is, under all heads, as the head of water.

The virtual head wherever the flow of water is regular, is proportional to the real head; hence for a given form of gate, the rule which

Remarks.

Table.

Feet.

is stated is general. When by varying the form we change the velocity of efflux, the effective head varies in the ratio of the square root of the velocity. In the case actually before us, the gate b is the one of the four forms used in different experiments which gave the greatest effective quantity of discharge.

It may be satisfactory to show that the head which would result from a calculation of the height due to the velocity of discharge, or virtual head, bore in these experiments a sensibly constant ratio to the real head, as has just been asserted.

The following table, containing the ratios of the observed velocities of the water, obtained from tables R, S, and T, (Jour. Frank. Inst. vol. ix, p. 154-7,) with the velocities calculated upon the heads above the aperture, will serve to show that these ratios vary within moderate limits, which are no doubt those within which the circumstances of the experiments may have varied without being detected.

TABLE THIRTY-FIRST.

Showing the ratio of the actual to the theoretical discharge under different heads. Gate b, undershot.

Head.

It is plain, from the manner in which these results vary, as well as from the amount of variation, that there is no sufficient ground to reject the actual heads in favour of heads calculated from the velocities of impulse.

The calculation of the effect by using the actual head, has, therefore, been preferred by the Committee, as more simple than the reference to a virtual head. An examination of table thirtieth will show that except in one case the ratio of effect to power is greater with a greater aperture. This does not arise from any difference in the actual velocity of the water compared with the velocity due to the head,

which might accidentally show some such relation. If the effects were corrected, by taking the virtual heads instead of the real heads, the ratio would rather be increased than diminished. It is necessary to recognise it as a true effect whatever may be its cause.

The truth of these remarks will be rendered more apparent by the following table, in which the actual heads and apertures are registered; the proportional discharges under the same head, but with varied apertures, are compared, and the effects as observed are also compared. Further to show that accidents in the velocity of discharge have not produced an apparent law, the virtual heads due to the velocities are calculated, and the ratios before obtained of effects are corrected according to these heads. A true comparison of effects under the same heads, but with different quantities of discharge, thence results.

TABLE THIRTY-SECOND.

Comparison of the relative effects under the same head, and with different quantities of water.

It will be observed in this table that only those experiments have been used in which the aperture was considerably varied. In the first set we have for an increase of 35 per cent. in the quantity of discharge, one of 66 per cent. in the absolute effect, and at a mean with an increase of quantity from 1 to 1.25; the effects increase from 1 to 1.42, or the increase of absolute effect is greater than that of quantity in the proportion of 1.11 to 1. We must admit, then, as a result of experiment, that an increase in the ratio of power to effect may be obtained by an increase in the quantity of water expended, at least under the circumstances in which the water was delivered to the wheel in these experiments. The physical cause of this effect is doubtless the same as that referred to in page 363, Vol. ii., new series, as producing a similar effect in the velocity of an overshot wheel.

By comparing the ratios of power and effect, corresponding to the table just given, it will be found that when the quantity is increased at a mean twenty-five per cent., the ratio of effect to power is increased but two per cent.

2. On the relative velocity of the wheel and water for an undershot wheel. The ratio of the velocity of the wheel to that of the water has been deduced in the following table, taken from tables R, S, and T.

TABLE THIRTY-THIRD.

Showing the relative velocity of the wheel and water in an under

This table gives for the mean ratio very nearly what was before obtained in the overshot wheel.

The ratio of the velocity of the wheel to that of the water is near ly nine-sixteenths.

The relative velocities in Smeaton's experiments varied between the proportions of one-third and one-half. Oliver Evans gives nearly two-thirds as the result of his practice. It is worthy of remark that Waring's Theory of Undershot Wheels gives one-half for the relative velocities of the wheel and water.*

A close examination of this table would seem to warrant the inference that as the heads increase, the ratio of the velocity of the wheel to that of the water decreases. This, however, we apprehend is not a fair deduction. The results in relation to the velocity of the Am. Philos. Trans.-old series. Vol. iii, p. 147.

wheel seem by no means so regular as those relating to the proportion of effect to power; but in general it will be seen that where the aperture is increased with a given head, and only the low heads admitted of any considerable increase without throwing too much water against the wheel, the ratio of the velocity of the wheel to that of the water is usually increased. It also appears by the tables that only the larger apertures were used under the low heads, hence the experiments are not precisely comparable, a large aperture giving a greater velocity than a small one would have done. Sometimes the actual velocities of the water, (deduced from the quantities discharged) appear to have increased by an increase in the aperture; but in general the reverse is the case, as will be seen by the following table. TABLE THIRTY-FOURTH.

Showing the velocity of the water with different apertures and under the same head.

In the first four experiments an increase of velocity attended an increase of opening, in the last two a very slight increase, and in all the others a decrease. As these irregularities run through the series, it would not be proper to base a law upon the mean result.

A comparison of the ratio which the actual discharge bore to the theoretical discharge, with different heads and openings, would not be more satisfactory. While it abundantly appears that an increase of quantity up to a certain point slightly increases the apparent velocity, there is not sufficient reason to deduce a law as to the increase of the latter element by an increase of the former; a conclusion entirely confirmed by general reasoning as to the effect produced by the physical cause to which these phenomena have been satisfactorily traced.

3. On the effect of air vents in undershot wheels.-At low heads