the progressive velocity of the earth's centre will be found to be no more, than that with which a body would traverse about of one millionth of one millionth

the progressive force which would be impressed on the earth, if all the force of each particle were effective. In what proportion the progressive force is diminished, on account of the various obliquities of impulse, is thus investigated.

K

O

T If ed B

Imagine T to be the earth's centre, and ACB to be half a great circle of the earth, perpendicular to that which separates the enlightened and the dark hemisphere, and which shall be called the terminator. On the plane of this semicircle, suppose the terminator, and its parallels in the enlightened hemisphere, to be projected, into right lines AB, kh, lm, no, &c. which are diameters of the circles respectively, of which they are the projections. The sun's distance may be considered as infinite; and therefore the rays of light, i. e. the directions of the particles, when they reach the earth's surface, are to be considered as parallel to each other, and all of them perpendicular to the plane of the terminator. Now imagine the whole enlightened hemisphere to be divided into innumerable little zones AB kh, hkml, mlno, &c. by small circles parallel to the terminator. Let the breadths of these little zones, measured on a great circle passing through the poles of the terminator, that is, let the infinitesimal arcs вk, km, no, &c. be so proportioned to each other, that perpen diculars kd, me, of, &c. being drawn from the extremities of these arcs, to the right line AB, which is the common intersection of the great circle ACB, and the plane of the terminator, the infinitesimal segments of that line вd, de, ef, &c. may be equal. Now imagine the particles of light which fall upon any one of these little zones, for instance, noqp, to meet with no resistance from the earth's surface, but to penetrate the globe, and to pass on without refraction or inflection, in the direction perpendicular to the terminator, till they arrive at the plane of the terminator, and there suppose them to stop, and each to lie still, in the place on which it falls. It is evident that the particles of light that fall upon, and have been supposed to pass through, the spherical zone pqon, will, with their proper interstices, cover that annular space on the plane of the terminator, which is the orthographical projection of the zone pqon, on that plane, and is comprised between the circumferences of circles, of which the right lines Tg and Tf are the radii. Hence the number of the particles of light, which fall upon the evanescent zone pqon, are as that evanescent annular space which they cover, that is, as gfx the circumference of the circle of which Tf is the radius, that is, as gf x in the right line f. But that part of the force of each particle, impinging on the zone pqon, which is perpendicular to the surface of the zone, is as of, if TB (the semidiameter of the earth) be put for the whole force. For join To, and draw fK perpendicular to To. The particle impinging at o moves in the direction of. Let the right line of then express its whole force, and this force of is composed of the two ok, Kf, of which ok is perpendicular to the surface of the sphere ato, and Kƒ is parallel to it. But ox: of of: OT or TB. Again, through к, draw xt perpendicular to of. The force ok is resolved into two ot, tk, of which ot is perpendicular to the plane of the terminator, and is the only part of the force OK, which tends to produce a progressive motion of the globe, in the direction of the impinging particles, that is, directly from the sun. The other part tк urges the centre of the globe along the plane of the terminator; but the forces tk being equal and contrary on opposite sides of, and at equal distances from the perpendicular ray, destroy each other's effects. Now it has been shown, that the whole force of the particle impinging at o, is to that part of its force which is perpendicular to the earth's surface at o, as TB to of. And it is manifest that ok is to ot, that is, the perpendicular force, of the particle impinging at o, is to that part of it which is effective in moving the earth's centre, as

"

of an inch in 100". This is the utmost effect of the force impressed on the earth by each emission. If the emissions were incessant, this might be considered as a central force, counteracting the sun's attraction; for its tendency is to push the earth directly from the sun. I need not say, that it is infinitely too small, in comparison of the sun's attraction, to produce any sensible effect.

The rotatory forces mentioned in the last note, if they were infinitely greater than they really are, would not in the least degree disturb the diurnal rotation; because every one of them is destroyed, by an equal one, in a contrary direction, on the other side of, and at an equal distance from, the perpendicular ray.

I have inquired, what may be the utmost stroke, which the retina of a common eye sustains, when the eye, in a bright day, is turned up directly to the sun. This force will evidently be at its maximum, if the emission be reckoned at its maximum. The number of particles which enter an eye, looking up directly at the sun, are to the number out of each emission which are directed towards the earth, in the duplicate proportion of the diameter of the pupil to the diameter of the earth. And the force with which the eye is struck, is to the sum of the forces of all the particles which strike the earth, in the same proportion. If therefore the diameter of the pupil, when the eye is exposed to the direct impulse of the sun's rays, be reckoned of an inch, which I apprehend must rather exceed than fall short of its real magnitude, in those circumstances, it will be found that every stroke which it receives from them, exceeds not that which an iron shot, of an inch diameter, would give, moving only at the rate of 16.16 ÷

TB to of. Therefore the whole force of the particle impinging at e, is to its effective part, as TB2 to of. That is, the effective part is as of. The number of particles impinging on the zone has been shown to be as gfx Tf. The progressive force of motion excited in the earth's centre, by all the particles impinging on the infinitesimal zone pqon, must be as the number of the particles and the effective part of each jointly; that is, as gf x Tfx of, or writing a for TB, and r for Bf, as xx (a — x) × (2ax-x2). And this is the fluxion of the progressive force of motion excited in the globe, by the particles impinging on that segment of the sphere, of which pABq is the projection. The number of particles impinging on the zone pqon, being as gf x Tf, or as x x (a — x), if each impinged perpendicularly, and its whole force were effective, the sum of the effective forces impressed on the whole, would be as gf x Tf x Tв2, or * × (a3 — a2x). And this would be the fluxion of the progressive force of motion of the globe, excited by the particles impinging on the segment of which pABq is the projection, if all impinged in directions perpendicular to the surface, and the whole of their forces were effective. The fluent of x × (a — x) × (2ax · x2) is 4 × (2ax — x2). And the fluent of * × (a3 — a2r) is a3x ar. When ra, the first of these two fluents is the sum of the progressive forces actually impressed on the whole hemisphere ACB, and the latter is the sum of the forces which would be so impressed, if all the impinging particles impinged perpendicularly, and the whole force of each were effective. But when a the first fluent becomes a4. And the latter becomes at. Whence it is manifest, that the progressive motion communicated to the globe of the earth, by the particles of light, is to the force which they would communicate, if the whole force of each were effective, in the proportion before assigned, of 1 to 2.-Orig.

inches in a year. This would be the stroke if the emission were at its maximum. Is it not owing to the extreme minuteness of the fibres of the nerves, that a stroke, which is certainly less than the part of this, is not sustained by our organs, without pain?

XXXVI. Some New Theorems for Computing the Areas of certain Curve Lines. By Mr. John Landen, F. R.S. p. 441.

or

;

The learned editor of Mr. Cotes's Harmonia Mensurarum first gave us, in that book, the celebrated theorems for computing the areas of the curves whose ordinates are expressed by and sea” + x”3 (a” + x") × (e" + x”) a2 + 2ca"x" + 202n veral other writers have since done the like. Which theorems consist of many terms, being obtained by previously resolving the expression for the ordinate, into others of a more simple form. Now I have found, says Mr. L., that the whole area of every such curve, when finite, may be assigned by theorems remarkably concise, without the trouble of resolving the expression for the ordinate as aforesaid; and as in the resolution of problems, the whole area of a curve is more commonly wanted than a part of it; and as these new theorems enable us to compute such whole areas as above mentioned, or the whole fluents of and with admirable facility; I do my

a” + x”' (a” + x”) × (e" + x”)' a2 + 2ca"x" + pang self the honour of communicating them to the Royal Society, presuming they may be thought worthy to be published in the Phil. Trans.

Theorem 1. m being any positive integer or fraction, and ʼn any such integer or fraction, greater than m; the whole area of the curve, whose abscissa is x, and ordinate

M8

is =

fn

ХА.

Theorem 2. m and n being as before mentioned, the whole area of the curve, whose abscissa is x, and ordinate

(a”+ x”) × (e” + 20%)

a" e±'

is = ±

A

X

an e"

fn

ma ±m-n

X A.

Note, when e is = a, the expression for the area becomes = fnz Theorem 3. m and n being as in the preceding theorems, the whole area of

Note. If m be= 0, the area will be =
In these theorems,

A denotes the semi-periphery of the circle, whose radius is 1;
B an arc of the same circle, whose cosine is c and sine b;

Concerning the investigation of these theorems, it is sufficient to say, they are

directly obtained by the help of my new method of comparing curvilineal areas, inserted in the Phil. Trans. for the year 1768.

It is obvious, that, by means of the above theorems, we may very readily compute the whole areas, when finite, of the curves, whose ordinates are

x

and

P+ qx" + rx2 + x3×9 p + qx" + rx2 + sx3" + xu &c. seeing these expressions maybe easily transformed into others similar to those already considered.

XXXVII. Transit of Venus observed in India. By Capt. Alexander Rose, of the 52d Regiment. Communicated by Dr. Murdoch, F.R.S. p. 444. Having procured a telescope and stop-watch, Capt. R. made observations on the transit of Venus, which happened on the 4th of June 1769.

Phesabad, lat. 25° 30′ north.

XXXVIII. Of a Periodical Fever, followed by a Separation of the Cuticle. By Mr. John Latham, Surgeon at Dartford, Kent. p. 451.

Mr. A. B., about 55 years of age, was a healthy man till about 20 years since, when he was first seized with a fever; at which time he followed the trade of a miller, and maker of French barley. This last business, he says, is attended with very great heat to the operator, and exposes him to a continual cloud of dust. As soon as he began to work, his breath became oppressed with a sensation of his body being puffed up all over; from which symptoms he was relieved by occasionally leaving off his business. On the first cold caught after his enter ing on this kind of employment, a fever attacked him; which has generally returned sometimes once, and sometimes twice in a year, chiefly in autumn; but sometimes in spring likewise: though he once missed being ill for 2 years to

Whence the planet's centre was on the sun's limb at 7h 1m 36'; and this compared with an observation of the central egress or ingress, made at a distant place, will give the sun's parallax; the other necessary elements of the calculus being well established. In the mean time we see, from the Connoisance des Tems for 1769, that Phesabad in Bengal, where Captain Rose observed, is 81° 45′ east of Paris.

The watch had been regulated the preceding day, by equal altitudes of the sun; the sun's altitudes, at the two contacts, are likewise marked in the captain's letter; but this part of the work he had probably entrusted to a less skilful observer, while his own attention was engrossed by the telescope and the watch; as I find the difference of the times correspondent, to those altitudes, does not agree with the interval of the contacts; for which reason they are here omitted.-P. MURDOCH. -Orig.

gether. After carrying on this trade for 4 or 5 years, he left it off; as he attributed his disorder chiefly to the effects of the meal dust. The fevers have not been so violent since, as while he followed that occupation: though the cuticle, or outer skin, has come off, the same as before. As to the particulars of his illness, they are nearly as follow: the disorder begins with a violent fever, attended with pains in the head, back, and limbs, accompanied with continual retchings; he sometimes vomited up much bile, at other times little or none; the skin was dry, the tongue much furred, together with great thirst, costiveness,, and the urine highly coloured. At the beginning of the fever he was generally let blood; this evacuation afforded some relief, and by keeping his body open, and taking cooling medicines, the retchings abated in about 5 or 6 days: the whole surface of the body became yellow, though this circumstance did not always happen. Afterwards it became florid, having the appearance of a rash; on which he felt a great uneasiness for several days, with a numbness and tingling all over him; when the urine became turned, and deposited a thick sediment. About the beginning of the 3d week from the first attack, the cuticle appeared elevated in many places. In 8 or 9 days afterwards it became so loose as to admit of being easily removed in large flakes. The cuticle of the hands from the wrist to the fingers' ends came off whole, bearing the resemblance of a glove. He never was disposed to sweat in any part of his illness, and when sweating was attempted by medicines he grew worse for it; nor was he much at ease till his urine deposited a sediment, after which he felt very little inconvenience, but from the rigidity of the skin. The nails of the patient, in a case communicated to the R. S., are mentioned to have come off after the illness; Mr. L. did not find that this was ever the case in this person.

Of a Very Small Foetus. By Mr. Joseph Warner. p. 453.

With the above cuticular glove was sent to the R. S. by Mr. Warner, a very small fœtus, brought into the world at the same time with a live child at its full growth. The woman was delivered before he came to her: on examining the placenta, a substance appeared somewhat unusual; and on washing it clean, he discovered the foetus above mentioned. It had no visible communication with: the placenta, but was squeezed flat, though not in the least putrid, and seemed shrivelled. He did not remember a case like this mentioned, except in Smellie's Midwifery, vol. 2, p. 85, where he relates one from the Academy of Sciences at Paris, nearly similar to this. May we not suppose the woman to have been with child of twins; and that this dying was not discharged, as was most likely to happen, but remained till the time of the natural birth, when they were both. expelled together?