On the Method of determining, from the real Probabilities of Life, the Values of Contingent Reversions in which three Lives are involved in the Survivorship. By William Morgan, Esq. F.R.S. Read Dec. 12, 1799. [Phil. Trans. 1800, p. 22.]

Mr. Morgan having already communicated to the Society the solutions of seventeen different problems in the doctrine of contingent reversions depending upon three lives, has been induced, from a wish to complete the subject, to investigate in the present paper seven more problems in which the same number of lives are concerned in the survivorship. These, he tells us, include, as far as he can perceive, all the remaining cases involving those complicated contingencies.

In examining the investigation of these problems, it appears that the determination of the reversion in some of them depends in each year on the happening of twelve or thirteen different events. These numerous contingencies being all expressed by separate fractions (each of which is resolved into two or more different series) renders the operations exceedingly intricate and laborious. From an apprehension, it seems, of becoming tedious and diffusive in his demonstrations, the author has in general contented himself with merely giving the fractions denoting the contingencies on which the reversion depends, without specifying in words at length the nature of those contingencies. He has, however, in these as in all the other problems he has investigated, given different demonstrations, both by solving each independent of any other problem, and by deriving the solution from those of two or more problems, which had been already investigated; so that from the exact agreement in the results, proofs are deduced of the perfect accuracy of the demonstration, not only of the problem investigated, but also of those which are applied to the solution.

In all these problems, a contingency is involved, which having never been accurately determined. had hitherto rendered even an approximation to the solution of them impossible. This contingency is that of one life's failing after another in a given time. This appears to have been ascertained with sufficient accuracy to enable the author to surmount a difficulty in the solution of these problems, which he owns he had once considered as insuperable.

Having thus accomplished the investigation of every case in which he conceives it possible that the contingency may be varied between these lives, he conceives that he has now exhausted the subject; and concludes his paper with observing, that those cases in which four lives are involved in the survivorship are not only too numerous and complicated to admit of solution, but that they occur so seldom in practice as to render the labour of such solution (if it were practicable) both useless and unnecessary.

Abstract of a Register of the Barometer, Thermometer, and Rain, at Lyndon, in Rutland, for the year 1798. By Thomas Barker, Esq. Read Dec. 12, 1799. [Phil. Trans. 1800, p. 46.]

On the Power of penetrating into Space by Telescopes; with a comparative Determination of the Extent of that Power in natural Vision, and in Telescopes of various Sizes and Constructions; illustrated by select Observations. By William Herschel, LL.D. F.R.S. Read Nov. 21, 1799. [Phil. Trans. 1800, p. 49.]

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It has long been observed that the power of distinguishing objects at great distances depends not only on the magnifying power applied to the telescope through which they are viewed, but also on the quantity of light emitted by the object, and collected and conveyed to the eye by means of the instrument. The superiority of telescopes with large apertures must hence appear obvious; and we have long witnessed the essential improvements made in this respect by Dr. Herschel, which have enabled him to extend his view into the firmament to distances, the bare mention of which is sufficient to astonish a mind unaccustomed to investigations of this nature. That it is principally the increased quantity of light that enables us to view luminous objects at great distances will appear manifest if we reflect that, since the density of light decreases in the ratio of the squares of the distances of the objects emitting the light, it follows that an object may be removed to distances at which its light will be so rarefied as to produce no longer any sensation upon the optic nerve that if an optical instrument be used with an object-glass of a larger diameter than the pupil of the eye, the quantity of light collected by this means in the eye will be greater in proportion to the greater extent of the object-glass compared with that of the pupil: and that hence the most distant star that can be seen with the naked eye, if it be viewed through a tube with an object-glass of twice the diameter of the pupil, it will without any magnifying power be visible at a distance four times greater than that at which the naked eye ceased to perceive it. Dr. Herschel many years ago adverted to this circumstance, when in his paper on the Construction of the Heavens, he introduced what he then figuratively called his sounding line, to which he now substitutes the appellation of the power of penetrating into space. And in the present paper he fully investigates a comparative determination of the extent of that power in natural vision, and in telescopes of various sizes and constructions; all which he illustrates by a number of select and curious observations.

In the first part of the paper he establishes the difference between magnifying and penetrating powers; he rejects some vague terms in common acceptation, to which he substitutes algebraic symbols and such accurate definitions as enable him to proceed upon solid ground. And after distinguishing between self-luminous objects and those which shine by a reflected light, and likewise noticing those whose

brightness is the effect of a considerable depth of luminous matter, he shows that these differences noways affect the present inquiry; since in all these several bodies, it is the quantity of light emitted by their surfaces which becomes the object of our perception. As the same body, however, may be differently luminous in different parts of its surface, he exhibits a formula by which the aggregate brightness of a body may be estimated. And he closes this part with an examination of the opinion maintained by Lambert in his Système du Monde, where he says that an object is equally bright at all distances, and that the sun at the distance of Saturn, or still further from us, would be as bright as it is in its present situation. This assertion taken in the general sense in which it is here expressed, he proves to be a palpable contradiction; and only admits it in as far as the celebrated author may mean the intrinsic brightness of a body, which applies to its surface diminished by distance, and not the absolute brightness of the whole.

In the next section the author endeavours to ascertain the general extent of vision with the naked eye. As to those bodies which shine with a reflected light, he asserts that none have yet been seen more distant than the Georgian planet: admitting this as the maximum, it must after all excite our admiration that borrowed light should be perceptible to our naked eye at the distance of no less than eighteen hundred millions of miles; especially if we consider that the light of the sun on that planet is above 368 times less intense than it is on our earth, and that probably two thirds of that diminished light is absorbed by the planet.

The range of natural vision, with respect to self-luminous objects, is incomparably more extended. Passing over the intermediate steps by which our author arrives at his conclusions, we shall simply mention his inference that no single star above nine or at most ten times more distant than Sirius can possibly be perceived by the naked eye; admitting, however, that an accumulation of stars will be perceptible at a far greater distance.

From the power of penetrating into space by naked vision, our author proceeds to that of telescopes. Here he first calculates, by a method recommended by Mr. Bouguer, the quantity of light absorbed and dissipated by the reflection of the mirror, and refraction of the eye-glasses; and he finds that a common Newtonian with three lenses loses about ths of the whole light it receives, and that in a telescope of his own construction with two lenses this loss amounts to somewhat less than ths. The Doctor now enters into a full investigation of the penetrating power of his several telescopes under all the various circumstances he could devise, and illustrates the whole by a great number of observations, which serve to confirm the inferences deduced by him. Here we learn that the penetrating power of his 20-feet reflector, applied to a single star, may extend as far as 612 times the distance of Sirius, and also that his large telescope, with a penetrating power of 192, will show a single star of the 1342nd magnitude.

In the next sections he shows that the penetrating and magnifying powers, so far from assisting each other, will often prove reciprocally detrimental, which he thinks may be explained by admitting that while the light collected is employed in magnifying an object, it cannot be exerted in giving penetrating power, to which perhaps ought to be added the detrimental effect of the magnifying power on the heterogeneous ingredients floating in the atmosphere. Whatever be the cause, the fact is proved by various observations.

Lastly, he shows that as we must not limit our vision within the sphere of the single stars, we must call the united lustre of the sidereal system to our aid in stretching forward into space. Supposing one of these clusters of 5000 stars to be at one of those immense distances to which only a 40-feet reflector can reach, he calculates that this distance will exceed at least 300,000 times that of the most remote fixed star visible to the naked eye. He concludes with a rough calculation how much time it would take to sweep the heavens with a penetrating power of such an immense extent; and finds that in this climate, with his 40-feet reflector, with a magnifying power of 1000, this operation for the whole sphere would take no less than 811 years.

A second Appendix to the improved Solution of a Problem in physical Astronomy, inserted in the Philosophical Transactions for the Year 1798, containing some further Remarks, and improved Formula for computing the Coefficients A and B ; by which the arithmetical Work is considerably shortened and facilitated. By the Rev. John Hellins, B.D.F.R.S. and Vicar of Potter's Pury in Northamptonshire. Read Dec. 12, 1799. [Phil. Trans. 1800, p. 86.]

This paper relates to an improved solution of a problem by which swiftly converging series are obtained, which are useful in computing the mutual perturbations of the planets; and contains some further remarks and improved formulæ for computing the coefficients, by which the arithmetical work is considerably shortened and facilitated.

Account of a Peculiarity in the Distribution of the Arteries sent to the Limbs of slow-moving Animals; together with some other similar Facts. In a Letter from Mr. Anthony Carlisle, Surgeon, to John Symmons, Esq. F.R.S. Read Jan. 9, 1800. [Phil. Trans. 1800, p. 98.]

This peculiarity was first observed in the axillary arteries and in the iliacs of the Lemur tardigradus, which at their entrance into the upper and lower limbs were found to be suddenly divided into a considerable number of equal-sized cylinders, which occasionally anastomosed with each other, and were regularly distributed on the muscles; whilst the arteries proceeding to the other parts of the body divided in the usual arborescent form.

Upon prosecuting this inquiry, it was found that the Bradypus

tridactylus, and in some measure also the didactylus, has a similar distribution of these arteries.

This peculiar disposition of the arteries in the limbs of these slowmoving quadrupeds, it is thought cannot but retard the velocity of the blood passing into the muscles of the limbs. Whence the well known sluggishness of the animals, to whom this configuration seems as yet peculiar, will perhaps be ultimately accounted for. Something similar has been observed in the carotid artery of the lion, which it is thought may be subservient to the long continued exertion of the muscles of his jaws, in holding a powerful animal for a length of time; and lastly, it is conjectured that the ruminating animals have a somewhat similar aplexus of arteries in the neck, which operates in retarding the velocity of the fluids in those parts.

Outlines of Experiments and Inquiries respecting Sound and Light. By Thomas Young, M.D. F.R.S. In a Letter to Edward Whitaker Gray, M.D. Sec. R.S. Read Jan. 16, 1800. [Phil. Trans. 1800, p. 106.]

We are here presented with a numerous set of experiments and observations, which the author does not deliver as a series calculated to elucidate any particular object, but rather as the results of the first steps of an investigation; which being of considerable magnitude, and not to be accomplished in a short period of time, are here brought forward in a detached form, in order to preserve them from oblivion, should any unforeseen circumstances prevent his continuing the pursuit. They are classed under sixteen different heads, of which the following are the titles, and some of the principal inductions.

1. Of the Quantity of Air discharged through an Aperture.—This was deduced from the quantity of pressure of water, on a body of air rushing through a small aperture at the end of a tube. The result was, that the quantity of air discharged by a given aperture is nearly in the subduplicate ratio of the pressure; and that the ratio of the expenditures by different apertures, with the same pressure, lies between the ratio of their diameters, and that of their areas.

2. Of the Direction and Velocity of a Stream of Air.-These were determined by the stream, produced by a known pressure, being made to impinge, in a perpendicular direction, against a white plate, on which a scale of equal parts was delineated, and which was thinly covered with a coloured liquid. The results were here inferred from the breadth of the surface of the plate laid bare by the stream.-The experiments being repeated at different distances between the orifice and the plate, the longitudinal form of the stream could be hence deduced, their sections being bounded by curves, the nature of which could be determined by their ordinates and abscissæ. The numerous results obtained in this manner are entered in various tables, and likewise illustrated by figures, in which the longitudinal and not the transverse sections are exhibited to the eye.

3. Ocular Evidence of the Nature of Sound.-This is produced by