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since the width has in general a constant ratio to the depth, or, in the case more particularly considered, since the width is equal to the depth, the quantity flowing per unit of time will, as in the preceding case, be proportional to the power of the depth; or we have

Flow over G K=Ch2 √h, where ß is constant.

Hence if be the flow, in units of volume per unit of time, over a unit of length in EF, we have

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By multiplying this by l we get the quantity flowing over the entire middle part E F per unit of time; and so, denoting that quantity by Q", we have

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Adding the expressions for Q' and Q" together, we get for the total flow in the whole notch, which we may denote by Q,

or

or

Bm-a

But

B constant

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is a constant; and let it be denoted by 2b; and instead of the we may, in order now to use English letters, put a. Then

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which is the desired formula for the flow of water in a rectangular notch with two end contractions.

This formula admits of easy modification to give a formula suitable for a notch with only one end contraction *, thus:—

Let the width of the notch with only one end contraction be denoted by L (as in fig. 14). Then conceive a notch twice as wide with two end contractions as shown in fig. 15. The flow in this double space will, by the formula last obtained (12), be seen to be = a(2L-2bh)h; and so if we put now Q to denote the flow in the notch under consideration (shown in fig. 14), which will be half the flow in fig. 15, we have for the notch with only one end contraction

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It is to be understood that contraction may be prevented at either end of a notch by there being a vertical plane side face for the channel of approach to the notch, that side face being perpendicular to the plane of the notch, and extending up-stream from the notch so as to reach beyond the region of incipient rapid flow to the notch, and extending for a little way down-stream past the notch, so as to afford the necessary guidance to the issuing stream-filaments. In like manner, by two parallel vertical side walls or side faces to the channel, when the crest of the notch extends quite across from the one wall-face to the other, contraction may be prevented at both ends.

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Also from (11), by changing, as done before, the letter 3 into the English letter a, we see that for a notch with no end contraction (contractions being prevented at both ends by vertical guiding side faces perpendicular to the plane of the notch) we would have

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Now the three formulas (12), (13), and (14) may be combined so as to be expressed together, thus :

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where n is the number of end contractions, and must be either 2, 1, or 0. To determine the constants a and b, all that would be necessary would be to have two very accurate experiments on the flow of water in one notch at different depths, or in two notches of the same kind with the ratio of the width to the depth not the same in both. Then, putting into the formula the measured values of L, h, and Q for the one experiment, and then again those for the other, we would have two equations with two unknown symbols, and so we could find the numerical values of those symbols. It would, of course, be desirable, for experimental verification of the theory on which the formula is founded, as also for mutual verification or testing of the experimental results themselves, to have numerous experiments on the flow for various depths in various notches of different widths, so as to find whether the formula would fit satisfactorily to them all, or to all of them that, after comparison, would be found trustworthy-provided that the width of the notch be not too small in proportion to the depth of the flow, or that in all cases the width be sufficient to allow of there being at least some small part in the middle where the rate of flow per unit of time would be proportional to the length of the part of the crest to which that flow would belong.

Mr. Francis's experiments and his reductions of the results carried out in

his own way give the formula complete, with its numerical coefficients, as follows*:

Q=3·33(L-nh)W3,

where Q the discharge in cubic feet per second;

L=the length of the notch in feet;

n the number of end contractions;

h=the height from the crest to the still-water surface-level in feet.

Mr. Francis also states that this formula is not applicable to cases in which the height h from the crest to the still-water surface-level exceeds one third of the length, nor to very small depths. In the experiments from which it was determined the depths varied from 7 inches to 19 inches; and he remarks that there seems no reason why it should not be applied with safety to any depths between 6 inches and 24 inches.

Report of the Anthropometric Committee, consisting of Dr. BEDDOE, Lord ABERDARE, Dr. FARR, Mr. FRANCIS GALTON, Sir HENRY RAWLINSON, Colonel LANE Fox, Sir RAWSON RAWSON, Mr. JAMES HEYWOOD, Dr. MOUAT, Professor ROLLESTON, Mr. HALLETT, Mr. FELLOWS, and Professor LEONE LEVI.

The Anthropometric Committee have been engaged during the past year in preparatory work. They have secured the cooperation of gentlemen holding positions under Government as inspectors of the army, of the navy, of factories, and of pauper schools. They have prepared schedules and instructions, and have had them printed; and they have purchased a small outfit of instruments to send to places where measurements are to be made in large numbers.

Under these circumstances they are unable to make a report of anthropometric results; neither have they been called upon to expend more than a small portion of the grant of £100 that was made to them in 1875, the larger part of which will be required to pay for the reduction of observations. Consequently they ask that the Anthropometric Committee may be reappointed, with modifications, and that the grant may be carried forward to the year 1876.

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On Cyclone and Rainfall Periodicities in connexion with the Sun-spot Periodicity. By CHARLES MELDRUM.

[Printed in extenso by the authority of the Council.]

In continuation of the paper on this subject published in the Report for 1874 (pp. 218-240), I beg to submit the following brief discussion of the cyclones of the Indian Ocean, between the equator and 34° S., in the years 1868-75, and of the rainfall in different places from 1854 to 1872.

Cyclones.

The number of cyclones in each year, the positions of their centres at noon on each day, their extent, duration, &c. have been approximately determined in the way already described, and the results are given in Table I.

From that and the similar Table given in 1874 we obtain the following general results for the twenty years 1856-75:

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It will be seen that, on the whole, the number of cyclones increased from 1857 to 1862, decreased from 1862 to 1867, then increased to 1870, and again decreased to 1875.

The distances traversed had nearly a similar progression, increasing from 1856 to 1861, decreasing from 1861 to 1867, then increasing to 1872, and again decreasing to 1875.

The areas have been determined by finding as nearly as possible the radii

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of the spaces (considered more or less circular) over which the wind blew with the force of a strong gale." They therefore are not the entire areas. But, apart from this, owing to incomplete information, the radii are not known for each day, and hence the areas are only rough approximations. There is no doubt, however, that they increased from 1856 or 1857 to 1860, decreased from 1860 to 1867, increased from 1867 to 1872, and then decreased to 1875.

On the whole, there was a similar progression in the duration of the cyclones, the smallest number of days being in 1856, 1857, 1867, and 1875, and the greatest in 1861 and 1870.

The total areas, i. e. the products of the mean area of each cyclone by the number of days it lasted, increased from 1856 to 1861, decreased from 1861 to 1867, increased from 1867 to 1872, and then decreased to 1875.

It is to be remarked, however, that the total areas for the years 1860-62 were much greater than those for the years 1870-72. This may be owing partly to the radii for the latter years having been underestimated. On the other hand, the number of cyclone-days in the years 1870-72 was somewhat greater than in the years 1859-61.

Rainfall.

A sufficient number of rainfall returns for the years 1873-75 have not yet been obtained; but the annual mean rainfalls at seventy-seven stations from 1854 to 1863, and at seventy-two stations from 1864 to 1872, are given in Table II., in which all the rainfall observations at my disposal have been used, except a few Prussian and Mauritius ones, which would not have affected the general results.

The Table shows that, with hardly an exception, the sun-spots and rainfall were both above or both below their respective averages in the same years.

By taking the longer period 1843-72, and expressing the amounts of rainfall and sun-spots in percentages, we get the following results:

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