« PreviousContinue »
13°; and the southern latitude of the perihelion also 13°: the ascending or north node 4o in Scorpio; and the comet's motion direct, or according to the order of the signs of the zodiac. On this supposition, he for some of the times of observations, estimated the apparent places of the comet, and found them as follows:
The observations which he took to ground the measurement on, are those of the 16th and 23d of September, and of the 1st of October. It appears very evident, not only from this rough calculation, but every other circumstance of this comet, that it is not the same with that in the year 1682, nor none of those already calculated or brought upon a list, by Messieurs Halley and Struyk. It is somewhat remarkable, that the line of the nodes is almost at right angles with the longer axis of the ellipsis; which corresponds nearly with the comets of the years 1580, 1683, and 1686: but those had their perihelions northward of the ecliptic; whereas the perihelion of the last, which we have lately seen, was to the southward of the ecliptic..
LXI. On the Different Temperature of the Air at Edystone, from that observed at Plymouth, between the 7th and 14th of July 1757. By Mr. John Smeaton, F. R. S. p. 488.
Edystone is distant from Plymouth about 16 miles, and without the headlands of the sound about 11. The 7th and 8th were not remarkable at Edystone for heat or cold: the weather was very moderate, with a light breeze at east; which allowed them to work upon the rock both days, when the tide served.
About midnight, between the 8th and 9th, the wind being then fresh at east, it was remarkably cold for the season. The wind continued cold the 9th all day; and so continued till the 10th; when seeing no prospect of a sudden alteration. of weather, Mr. S. returned to Plymouth in a sailing boat, wrapped up in his thick coat. As soon as they got within the headlands, he could perceive the wind to blow considerably warmer; but not so warm as to make his great coat uneasy. Having had a quick passage, in this manner he went home, to the great astonishment of the family to see him so wrapped up, when they were complaining of the excessive heat: and indeed, it was not long before he had reason to join in their opinion..
This heat he experienced till the 12th, when he again went off to sea, where he found the air very temperate, rather cool than warm; and so continued till the 14th.
It may not be amiss further to observe on this head, that once in returning from Edystone, having got within about 2 miles of the Ramhead, they were becalmed; and here they rolled about for at least 4 hours; and yet at the same time saw vessels not above a league from them, going out of Plymouth Sound with a fresh of wind, the direction of which was towards them, as they could observe from the trim of their sails; and as they themselves experienced, after they got into it by tacking and rowing.
Hence it appears, how different the temperature of the air may be in a small distance; and to what small spaces squalls of wind are sometimes confined.
LXII. Of the Earthquake felt in the Island of Sumatra, in the East Indies, in Nov. and Dec. 1756. In a Letter from Mr. Perry, at Fort Marlborough, Feb. 20, 1757. Communicated by the Rev. William Stukeley, M. D., F.R. S. p. 491.
The earthquake at Lisbon was certainly one of the most awful and tremendous calamities that has ever happened in the world. Its effects are extremely wonderful and amazing; and it seems to have been felt in all parts of the globe. On the 3d day of the same month the earthquake of Lisbon happened, Mr. P. himself felt at Manna * a violent shock; and from that time to the 3d of December following he felt no less than 12 different shocks. Since which we have had 2 very severe earthquakes, felt we believe throughout this island. The walls of Cumberland-house were greatly damaged by them. Several houses, the houses of Laye|| and Manna, were all cracked by them; and the works at the sugar-plantation § received considerable damage. The ground opened near the qualloe at Bencoolen, and up the river in several places; from which issued sulphureous earth, and great quantities of water, with a most intolerable stench. Poblo Point ** as much cracked at the same time; and some doo soons in-land at Manna were destroyed, and many people in them.
* Manna lies about 50 miles to the southward of Marlborough.-Orig.
↑ The island of Sumatra is between 7 and 8 hundred miles long, from north to south.-Orig. Cumberland house is a new well-built house for the governor of the place.-Orig.
Laye house or factory is about 30 miles to the northward of Marlborough, and Manna house or
factory 50 miles to the southward.-Orig.
The sugar plantation is 5 or 6 miles from Marlborough.-Orig.
¶ The qualloe is the country word for a river's mouth.-Orig.
** Poblo Point lies about 3 leagues to the southward of Marlborough,—Orig.
++ Doosoons are villages.-Orig.
LXIII. Concerning the Fall of Water under Bridges. By Mr. J. Robertson. F. R. S. p. 492.
Some time before the year 1740, the problem about the fall of water, occasioned by the piers of bridges built across a river, was much spoken of at London, on account of the fall that it was supposed would be at the new bridge to be built at Westminster. In Mr. Hawksmore's and Mr. Labelye's pamphlets, the former published in 1736, and the latter in 1739, the result of Mr. Labelye's computations was given: but neither the investigation of the problem, nor any rules, were at that time exhibited to the public.
In the year 1742 was published Gardiner's edition of Vlacq's Tables; in which among the examples there prefixed to show some of the uses of those tables drawn up by the late William Jones, Esq. there are 2 examples, one showing how to compute the fall of water at London bridge, and the other applied to Westminster bridge: but that excellent mathematician's investigation of the rule, by which those examples were wrought, was not printed, though he communicated copies of it to several of his friends. Since that time, it seems as if the problem had in general been forgotten, as it has not made its appearance in any of the subsequent publications. As it is a problem somewhat curious, though not difficult, and its solution not generally known (having seen 4 different solutions, one of them very imperfect, extracted from the private books of an office in one of the departments of engineering in a neighbouring nation,) Mr. R. thought it might give some entertainment to the curious in these matters, if the whole process were published. In the following investigation, much the same with Mr. Jones's, as the demonstrations of the principles used appeared to be wanting, they are here attempted to be supplied.
1. A heavy body, that in the first second of time has fallen the height of a .eet, has acquired such a velocity, that, moving uniformly with it, will in the next second of time move the length of 2a feet.
2. The spaces run through by falling bodies are proportional to one another as the squares of their last or acquired velocities.
These two principles are demonstrated by the writers on mechanics.
3. Water forced out of a larger channel through one or more smaller passages, will have the streams through those passages contracted in the ratio of 25 to 21. This is shown in the 36th prop. of the 2d book of Newton's Principia.
4. In any stream of water, the velocity is such, as would be acquired by the fall of a body from a height above the surface of that stream.
This is evident from the nature of motion.
5. The velocities of water through different passages of the same height, are reciprocally proportional to their breadths.-For, at some time the water must be delivered as fast as it comes; otherswise the bounds would be overflowed.. At that time, the same quantity which in any time flows through a section in the open channel, is delivered in equal time through the narrower passages; or the momentum in the narrower passages must be equal to the momentum in the open channel; or the rectangle under the section of the narrower passages, by their mean velocity, must be equal to the rectangle under the section of the open channel by its mean velocity. Therefore the velocity in the open channel, is to the velocity in the narrower passages, as the section of those passages, is to the section of the open channel. But the heights in both sections being equal, the sections are directly as the breadths; consequently the velocities are reciprocally as the breadths.
6. In a running stream, the water above any obstacles put in it, will rise to such a height, that by its fall the stream may be discharged as fast as it comes.For the same body of water, which flowed in the open channel, must pass through the passages made by the obstacles: and the narrower the passages, the swifter will be the velocity of the water: but the swifter the velocity of the water, the greater is the height from which it has descended: consequently the obstacles which contract the channel, cause the water to rise against them. But the rise will cease when the water can run off as fast as it comes: and this must happen when, by the fall between the obstacles, the water will acquire a velocity in a reciprocal proportion to that in the open channel, as the breadth of the open channel is to the breadth of the narrow passages.
7. The quantity of the fall, caused by an obstacle in a running stream, is measured by the difference between the heights fallen from to acquire the velocities in the narrow passages and open channel.-For just above the fall, the velocity of the stream is such, as would be acquired by a body falling from a height higher than the surface of the water: and at the fall, the velocity of the stream is such, as would be acquired by the fall of a body from a height more elevated than the top of the falling stream: consequently the real fall is less than this height. Now as the stream comes to the fall with a velocity belonging to a fall above its surface; consequently the height belonging to the velocity at the fall, must be diminished by the height belonging to the velocity with which the stream arrives at the fall.
PROBLEM.-In a channel of running water, whose breadth is contracted by one or more obstacles; the breadth of the channel, the mean velocity of the whole stream, and the breadth of the water-way between the obstacles being given; to find the quantity of the fall occasioned by those obstacles.
Let b breadth of the channel in feet.
c = breadth of the water-way between the obstacles.
v = mean velocity of the water in feet per sec.
Now 25 21 cc the water-way contracted; by princip. 3.
v the veloc. per sec. in the water-way between the ob
And (2 a)2 : (255)1 × vv :: a: 1 (2a)2
v; by 1 and 2.
is the measure of the fall required; by 7.
is a rule, by which the fall is readily computed.
Here a = 16,0899 feet, and 4a = 64,3596.
EXAMPLE I. For London Bridge.
By the observations made by Mr. Labelye in 1746,
The breadth of the Thames at London bridge is 926 feet.
The sum of the water-ways at the time of the greatest fall is 236 feet.
The mean velocity of the stream taken at its surface just above bridge is 3 feet
Under almost all the arches there are great numbers of drip-shot piles, or piles driven into the bed of the water-way, to prevent it from being washed away by the fall. These drip-shot piles considerably contract the water-ways, at least of their measured breadth, or about 39+ feet in the whole.
So that the water-way will be reduced to 196 feet.
Now b = 926; c = 1963; v = 34; 4 a = 64,3596.
And 5,605322 = 31,4196; and 31,4196 — 1 — 30,4196 =
4 a 36 × 64,3596
Then 30,4196 x 0,15581 = 4,739 feet, the fall sought after.
By the most exact observations made about the year 1736, the measure of the fall was 4 feet 9 inches.
EXAMPLE II. For Westminster Bridge.
Though the breadth of the river at Westminster bridge is 1220 feet; yet, at the time of the greatest fall, there is water through only the 13 large arches, which amount to 820 feet: to which adding the breadth of the 12 intermediate piers, equal to 174 feet, gives 994 for the breadth of the river at that time: and