Philosophical transactions, Vol. L. Part II. For the year 1758.

_Longit._ _Latit._ 1757. ° ´ ° ´ Sept. 16. at 4 h. ante mer. The comet in ♋ 10 15 with 10 10 North. 17 -- 3 -- -- -- -- -- ♋ 14 7 ---- 9 38 18 -- 3¾ -- -- -- -- -- ♋ 18 10 ---- 8 57 19 -- 4 -- -- -- -- -- ♋ 22 1 ---- 8 17 22 -- 2¾ -- -- -- -- -- ♌ 3 46 ---- 6 15 23 -- 4 -- -- -- -- -- ♌ 7 36 ---- 5 24 25 -- 4¼ -- -- -- -- -- ♌ 14 50 ---- 4 6 28 -- 4 -- -- -- -- -- ♌ 24 22 ---- 1 41 Oct. 1 -- 4¾ -- -- -- -- -- ♍ 2 46 ---- 0 12 South. 4 -- 4½ -- -- -- -- -- ♍ 9 45 ---- 1 30 9 -- 4½ -- -- -- -- -- ♍ 20 20 ---- 2 40 11 -- 5 -- -- -- -- -- ♍ 24 46 ---- 3 9

But the two last observations will, in my opinion, differ the most; because, when I made them, I was in some doubt about the adjustment of my instruments; and the comet was then far advanced into the morning rays. I have, since the month of February last to the end of May, made sundry observations on fixed stars, with a telescope of 16 inches, made by Mr. Short; and with a pendulum clock, made after the manner of Mr. Graham, by Mr. Vryhthoff of this place. In the months of February and March, by a medium of eight observations, I found, that by the clock, the star Rigel, in every daily revolution, passed 4 min. 2⁴⁄₉ seconds of time earlier, in the telescope; and in the latter end of May I found, by six observations, (the clock not in the least changed or altered) on the star Spica Virginis, that that star, in every revolution, passed 4 min. 5¹⁄₂₀ sec. earlier, in the same telescope; which intervals differ pretty nearly 2⅗ seconds of time from one another. Whether this difference arises from any defect in the clock, or whether it proceeds from any small difference of velocity of the earth’s motion round its axis, I would have been very glad to have endeavoured to find out by farther inquiry, had not the death of Mr. S. Koenig intervened, and I thereby hindered from continuing my observations. The above observations were taken in the observatory of his illustrious Highness the minor Prince of Orange and Nassau, _&c. &c._ under the direction, and with the approbation of the aforesaid Mr. Koenig. After the death of that gentleman, I petitioned her Royal Highness the Princess Governess of these Provinces, _&c._ that I might have leave to continue my astronomical observations; but as yet I have not been able to obtain her Royal Highness’s permission: otherwise I would have observed this last comet with more exactness. Had I been able to pursue the above-mentioned observations, I would, for the greater certainty in regard to the pendulum, have made use of a farther precaution. By means of a stove, with the help of a thermometer, I would have endeavoured to have kept the room (in which the clock stood) in the winter, and at all times, in the same degree of heat it had at the time I made the observations in the summer. I would also have daily observed and noted the moon’s place, at the time of the observations. Tho’ this is but a slight observation of mine; yet I make no doubt, but that in case, by the different distances of the earth from the sun, and the different distances and situations of the moon with respect to the earth, and the respective effects produced by these causes, any inequality arises in the velocity of the diurnal motion of the earth on its axis, you (who have made the most sublime observations on the aberration of the fixed stars, and more than any mortal ever did before) must have discovered, and are well acquainted, with the same.

As my above-mentioned observations on the comet appeared too incorrect to undertake a calculation for the ascertaining of its path from the theory, I contented myself with effecting it by a construction. By this means I found, on a figure, whose globular or spherical diameter was 13½ Rhineland inches, as follows:

That the comet was in its perihelion the 21st of October, at two of the clock in the afternoon: the place of the perihelion 3 degrees in Leo. The comet’s distance in the perihelion from the sun was about 34 parts, whereof 100 make the mean distance between the sun and the earth. The inclination of the comet’s orbit with the ecliptic 13 degrees; and the southern latitude of the perihelion also 13 decrees: the ascending or north node ☊ 4⅓ degrees in Scorpio; and the comet’s motion direct, or according to the order of the signs of the zodiac. On this supposition I have, for some of the times of observations, estimated the apparent places of the comet, and found them as follows:

_Long._ _Latit._ Sept. 18, at 3¾ ante merid. In ♋ 18¹⁄₁₂ and 9 deg. North. 19 -- 4 -- -- -- ♋ 22 ---- 8⅖ 22 -- 2¾ -- -- -- ♌ 3⅝ ---- 6¼ 23 -- 4 -- -- -- ♌ 7⅗ ---- 5½ 25 -- 4¼ -- -- -- ♌ 14⅔ ---- 4 28 -- 4 -- -- -- ♌ 24⅓ ---- 1¾ Oct. 4 -- 4½ -- -- -- ♍ 9⅓ ---- 2 ---- South. 9 -- 4¾ -- -- -- ♍ 19⅔ ---- 3⅖ 11 -- 5 -- -- -- ♍ 23⅛ ---- 3⅘

The observations, which I have taken, 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: which, on certain accounts, is very desirable to be known; for both here, and in other parts of the Netherlands, there have been some people, who have published mere conjectures; and have ventured (very minutely and exactly, as they pretended) about the time that this comet first made its appearance, to predict the return of the comet of the year 1682. But by the above, the weakness of their pretensions is very evident to all the world: whereas, if this had proved to be the expected comet, they would have assumed to themselves much undue praise, and have pretended to knowlege even superior to the every-where much celebrated Newton and Halley.

It appears also probable to me, that this comet is 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 long 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.

I have the honour to subscribe myself, with the most perfect esteem for you, and your sublime studies, very respectfully,

SIR, Your very humble and obedient Servant, D. Klinkenberg.

Hague, 13th Dec. 1757.

LXI. _Remarks 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._

[Read Jan. 12, 1758.]

SIR,

ON the reading of Dr. Huxham’s letter at the last meeting, some observations occurred to me, concerning the different temperature of the air, which I had observed at the Edystone, from what had been observed by the Doctor at Plymouth, between the 7th and 14th of July last: which having been desired by some members to be put into writing, I beg leave to trouble you with the following.

Edystone is distant from Plymouth about 16 miles, and without the head-lands 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 us 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, as I had more particular occasion to observe, on account of a ship that was cast away upon the rocks. The wind continued cold the 9th all day; which was complained of by some of the shipwrecked seamen, who had not time to save their cloaths; and so fresh at east, as prevented our going near the rocks, or the wreck; and so continued till Sunday the 10th; when, seeing no prospect of a sudden alteration of weather, I returned to Plymouth in a sailing boat, wrapped up in my thick coat. As soon as we got within the headlands, I could perceive the wind to blow considerably warmer; but not so warm as to make my great coat uneasy. Having had a quick passage, in this manner I went home, to the great astonishment of the family to see me so wrapped up, when they were complaining of the excessive heat: and indeed, it was not long before I had reason to join in their opinion.

This heat I experienced till Tuesday the 12th, when I again went off to sea, where I found the air very temperate, rather cool than warm; and so continued till Thursday the 14th.

In my journal for Wednesday the 13th I find the following remarks, _viz._ “This evening’s tide” (from 6 A. till 12 A.) “the wind at east, but moderate, with frequent flashes of lightning to the southward. Soon after we got on board the store-vessel, a squall of wind arose from the south-west on a sudden, and continued for about a minute; part of which time it blew so hard, we expected the masts to go by the board: after which it was perfectly calm, and presently after a breeze returned from the east.”

And in the journal of the 14th is entered, “This morning’s tide” (_viz._ from 1 M. to 1 A.) “the air and sea quite calm.”

Hence it appears, how different the temper of the air may be in a small distance; and to what small spaces squalls of wind are sometimes confined.

It may not be amiss further to observe upon this head, that once, in returning from Edystone, having got within about two miles of the Ramhead, we were becalmed; and here we rolled about for at least four hours; and yet at the same time saw vessels, not above a league from us, going out of Plymouth Sound with a fresh of wind, whose direction was towards us, as we could observe from the trim of their sails; and as we ourselves experienced, after we got into it by tacking and rowing.

I am, Sir,

Your most humble Servant, J. Smeaton.

Furnival’s-Inn Court, 12th Jan. 1758.

LXII. _An Account of the Earthquake felt in the Island of_ Sumatra, _in the_ East-Indies, _in_ November _and_ December 1756. _In a Letter from Mr._ Perry _to the Rev. Dr._ Stukeley, _dated at_ Fort Marlborough, _in the Island of_ Sumatra, Feb. 20. 1757. _Communicated by the Rev._ Wm. Stukeley, _M. D. F.R.S._

[Read Jan. 12, 1758.]

THE earthquake at Lisbon, which you gave me an account of, was certainly the most awful tremendous calamity, that has ever happened in the world. Its effects are extremely wonderful and amazing; and it seems, as you observe, to have been felt in all parts of the globe. On the 3d day of the same month the earthquake of Lisbon happened, I felt at Manna[1] a violent shock myself; and from that time to the 3d of December following I felt no less than twelve different shocks, all which I took an exact account of in my pocket-bock. Since which we have had two very severe earthquakes, felt, we believe, throughout this island[2]. The walls of[3] Cumberland-house[4] were greatly damaged by them. Salop-house[4], my own (formerly Mr. Massey’s), the houses of Laye[5] and Manna, were all cracked by them; and the works at the sugar-plantation[6] received considerable damage. The ground opened near the _qualloe_[7] at Bencoolen, and up the River in several places; and there issued therefrom sulphureous earth, and large quantities of water, sending forth a most intolerable stench. Poblo Point[8] was much cracked at the same time; and some _doosoons_[9] in-land at Manna were destroyed, and many people in them.

These are all the ill effects, that have come to our knowlege; but, it is reasonable to suppose, not all the damage, that has happened upon the island.

LXIII. _Concerning the Fall of Water under Bridges. By Mr._ J. Robertson, _F.R.S._

[Read Jan. 19, 1758.]

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 talked 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 shew some of the uses of those tables drawn up by the late William Jones, Esq; there are two examples, one shewing 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, altho’ he communicated to several of his friends copies thereof. Since that time, it seems as if the problem had in general been forgot, as it has not made its appearance, to my knowlege, in any of the subsequent publications. As it is a problem somewhat curious, tho’ not difficult, and its solution not generally known (having seen four 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), I 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 therein used appeared to be wanting, they are here attempted to be supplied.

PRINCIPLES.

I. _A heavy body, that in the first second of time has fallen the height of a feet, has acquired such a velocity, that, moving uniformly therewith, will in the next second of time move the length of 2 a feet._

II. _The spaces run thro’ 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.

III. _Water forced out of a larger chanel thro’ one or more smaller passages, will have the streams thro’ those passages contracted in the ratio of 25 to 21._

This is shewn in the 36th prop. of the 2d book of Newton’s Principia.

IV. _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.

V. _The velocities of water thro’ 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; otherwise the bounds would be overflowed.

At that time, the same quantity, which in any time flows thro’ a section in the open chanel, is delivered in equal time thro’ the narrower passages; or the momentum in the narrow passages must be equal to the momentum in the open chanel; or the rectangle under the section of the narrow passages, by their mean velocity, must be equal to the rectangle under the section of the open chanel by its mean velocity.

Therefore the velocity in the open chanel is to the velocity in the narrower passages, as the section of those passages is to the section of the open chanel.

But the heights in both sections being equal, the sections are directly as the breadths;

Consequently the velocities are reciprocally as the breadths.

VI. _In a running stream, the water above any obstacles put therein 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 chanel, must pass thro’ 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 whence it has descended:

Consequently the obstacles, which contract the chanel, 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 chanel as the breadth of the open chanel is to the breadth of the narrow passages.

VII. _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 chanel._

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; and 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 chanel of running water, whose breadth is contracted by one or more obstacles; the breadth of the chanel, 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 chanel in feet. _v_ = mean velocity of the water in feet per sec. _c_ = breadth of the water-way between the obstacles.

Now 25: 21∷ _c_: 21 ⁄ 25 _c_ the water-way contracted. _Principle III._

And 21 ⁄ 25 _c_: _b_∷ _v_: 25_b_ ⁄ 21_c_ _v_ the veloc. _per_ sec. in the water-way between the obstacles. _Princip. V._

Also (2_a_)²: _vv_∷ _a_: _vv_ ⁄ 4_a_ the height fallen to acquire the vel. v. I. & II.

And (2_a_)²: (25_b_ ⁄ 21_c_)² × _vv_∷ _a_: (25_b_ ⁄ 21_c_)² × _vv_ ⁄ 4_a_ the height fallen to acquire the vel. 25_b_ ⁄ 21_c_ _v_. I. & II.

Then (25_b_ ⁄ 21_c_)² x (_vv_ ⁄ 4_a_) - (_vv_ ⁄ 4_a_) is the measure of the fall required. VII.

Or ((25_b_ ⁄ 21_c_)² - 1) × _vv_ ⁄ 4_a_ is a rule, by which the fall may be readily computed.

Here _a_ = 16,0899 feet and 4_a_ = 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 3⅙ feet _per_ second.

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 ⅙ of their measured breadth, or about 39⅓ feet in the whole.

So that the water-way will be reduced to 196⅔ feet.

Now _b_ = 926; _c_ = 196⅔; _v_ = 3⅙; 4_a_ = 64,3596.

Then 25_b_ ⁄ 21_c_ = 23150 ⁄ 4130 = 5,60532.

And 5,60532² = 31,4196; and 31,4196 - 1 = 30,4196 = (25_b_ ⁄ 21_c_)² - 1.

Also _vv_ = (19 ⁄ 6)² = 361 ⁄ 36; And _vv_ ⁄ 4_a_ = 361 ⁄ (36 × 64,3596) = 0,15581.

Then 30,4196 × 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._