fical aftronomy. The Doctor has added two very general theorems, of confiderable ufe in the doctrine of chances; and he concludes his paper by pointing out fome of the purposes to which they may be applied.

A Meteorological Journal, principally relating to Atmospheric Electricity; kept at Knightsbridge, from May 9, 1789, to May 8, 1790. By Mr. John Read.

This paper contains fome curious obfervations, and a defcription of the inftrument, ufed in the journal, for collecting atmospheric electricity.

The Longitudes of Dunkirk ana Paris, from Greenwich, deduced from the triangular Measurement in 1787 and 1788, fuppofing the Earth to be an Ellipfoid. By Mr. Ifaac Dalby. In the account of General Roy's trigonometrical operations in 1787 and 1788, given in the Phil. Tranf. vol. lxxx. for 1790, and of which an account is alfo given at page 406 of our Review for April 1791*, the distance of Dunkirk from the meridian of Greenwich was firft determined, on a line parallel to one fuppofed to be drawn perpendicular to the meridian of Greenwich, at the point where the obfervatory is fituated; after which the longitude of Dunkirk, from Greenwich, was found by fpherical computation, on a fuppofition that the furface of the fphere, to which these computations were adapted, coincided, very nearly, with that of the earth, in an eaft and weft direction, at the place where the operation was performed; and that the magnitude of this fphere had been determined by actual obfervations, made at two ftations, nearly in the latitude of Dunkirk. As the principles on which this computation was founded, though not strictly geometrical, call for no material objections, (the only hypothefis, that is admitted, being evidently fuch as can not affect the conclufions drawn from them in any fenfible degree, and as great care was taken in obferving the angles, at thofe ftations on which the direction of the meridians depend,) Mr. Dalby thinks that the longitude of Dunkirk, and, confequently, that of Paris, which is deduced by means of it, must be very near the truth, whatever the real figure of the earth may be.

As it may, however, be thought by fome, that the arc between the meridians of Botley Hill and Goudhurst, which is that determined by the obfervations, is too fhort to afcertain the value of the arc between the meridians of Greenwich and Dunkirk, with fufficient accuracy, as the former is but 17 miles, and the latter almost a degree and an half; and more efpecially as it is remarked, in the Appendix to the Soth vol. See likewile the fecond article in our last Appendix, just publifhed.

of

of the Philofophical Tranfactions, that an error of a single fecond in the determination of the former arc will create an error of nearly 6 feconds in the latter, Mr. Dalby has been induced to try what refult a different mode of calculation, founded on different principles, will bring out.

The principles, on which Mr. Dalby's computation is founded, are undoubtedly ftrictly geometrical: but it must not be thence inferred that his conclufion is more exact than that which is given in General Roy's paper; for his hypothefis, namely, that the earth is an ellipfoid, and which enables him to compute from ftrict geometrical principles, is by no means a certain one-there is, indeed, good reafon for thinking it falfe, though to what extent that falfehood may operate in any place, is uncertain; it may be as near the truth as M. Bouguer's hypothefis, on which the General's calculation is founded; or it may be much more erroneous: neither is his method of determining the magnitude of his ellipfoid unexceptionable; though, for any thing we know, it may be as unexceptionable as any other that he could have used. To explain this:-It is well known, to moft perfons who are converfant in these matters, that the feveral measures of a degree, which have been made in different latitudes, do not, by any means, agree with one another, if we suppose the figure of the earth to be that of a perfect spheroid; and, of course, the meridians to be perfect ellipfes. This dif agreement, it is alfo well known, may arife either from inaccuracy in the terreftrial measurements, from inaccuracy in determining the quantity of the celeftial arch which corresponds to those measurements, or from the figure of the earth not being that of a perfect spheroid: or the difagreement may arise (and moft probably does,) from a combination of all these three fources of error *. General Roy, in forming his calculation of the difference

Will not our readers wonder why fo much labour and coft has been expended in procuring data, which, after all, cannot be applied to the refolution of the problem under confideration, without the affumption of vague and indeterminate hypotheses; when a direct folution, void of all hypothefis whatever, prefents itself by means of aftronomical obfervations? As we profefs ourselves no "niggards of advice," fo neither would we be thought niggards of information, when we have any to give: we fhall therefore inform those who do not already know it, that in the year 1783 Comte d'Adbémar, the French Ambaffador, tranfmitted to Mr. Fox, then Secretary of State, a memoir of the late M. Caffini de Thury, in which he afferted that the longitude of the Royal Obfervatory, at Greenwich, (from Paris,) is not known, certainly, within leis than about 11"; and that its latitude is uncertain to the amount of 15"; and he recommended the latitude and longitude of it to be de

termined,

difference of longitude between Greenwich and Paris, fuppofed the difagreement to arife wholly from the curve of the meridian not being a perfect ellipfis: Mr. Dalby has fuppofed it to lie wholly in the measurements; and he thinks it confifts principally in the determinations of the celeftial arcs; for there is, fays he, little doubt but the terreftrial measurements are, in general, exact enough.

Whether the error lie in the terreftrial, or celeftial ́meafurements, is not material to the point under confideration: but it is very material to know, if it were poffible, which of the measured degrees is erroneous; if, as Mr. Dalby's hypothefis fuppofes, the error lies there; this, however, can only be conjectured, from the accounts which are given of the inftruments ufed, and the means employed, in making thofe measurements. Mr. Dalby finds that, if he fuppofes the measurements of a degree at the equator, by M. Condamine and his affociates, and at the polar circle, by M. Maupertuis and his co-operators, to be adopted as true, and if the meridians be fuppofed ellipfes, the length of a degree, in the latitude of 50° 9, which is the middle point between the parallels of Greenwich and Paris, will be greater, by 140 fathoms, than the measurement, performed jointly by General Roy and M. Caffini de Thury, makes it; and, as these two measurements of M. M. Condamine and Maupertuis were made in fituations which are better adapted to the purpose of determining the fpecies of the curve which forms the terrestrial meridians, fuppofing it to be an ellipfis of any kind, than any others are, it might have been fuppofed that Mr. Dalby would have affumed that fpecies of ellipfis which refults from these two meafurements:-but, inftead of doing fo, he has fuppofed both thefe measurements to be erroneous in excefs, to the amount of 140 fathoms, or nearly fo; and, taking the measured degree in latitude 50" 9 N. to be 60844 fathoms, as the late operation by General Roy and M. Caffini makes it, and that of M. Condamine, at the equator, to be only 60344, or 137 fathoms less than M. Condamine gave it; he determines the termined, by terrestrial measurement, from the Obfervatory at Paris. To all those who were acquainted with aftronomical obfervations, and particularly with the obfervations which have been made at Greenwich, this French memorial appeared to be unworthy of confideration: but fome, who were not fo well acquainted with the merits of it, thought otherwife; and it was fuggefted, that an attempt to afcertain the point in queftion that way, might lay the foundation of, and be the means of fetting on foot, a general furvey of the whole kingdom: his Majefty, ever ready to promote the extenfion of fcience, furnished the money which was neccffary for carrying it on, from his privy purfe.

fpecies

fpecies and magnitude of an elliptical meridian, which will agree nearly with all the degrees that have been measured in the middle latitudes. Had the measurement of a degree of the meridian between Greenwich and Paris been the only one which difagreed with the fpecies and magnitude of the terrestrial meridian, deducible from the meafurements of M. M. Condamine and Maupertuis, we should not have fcrupled to exprefs our difapprobation of Mr. Dalby's conduct in this bufinefs: but it is remarkable that the ellipfis deduced from those two celebrated measurements, will not agree with any of the measurements which have been made in the middle latitudes; and, therefore, if Mr. Dalby calculated at all on the hypothefis of the earth being a fpheroid, we think there was fome reafon for rejecting that fpecies of ellipfis which results from these measurements, and for recurring to one which agrees better with the obfervations that have been made in latitudes that correspond more nearly with that for which his computations were to be made; and, calculating from thofe principles, Mr. Dalby finds that the difference of longitude between Greenwich and Paris is 9' 20", or only about 1" greater than the determination in General Roy's paper.

Mr. Dalby fays, if, inftead of fuppofing the meafurement at the equator and polar circle to be too great by 140 fathoms, we fuppofe each of them too great by only 70 fathoms; and, of course, the length of the degree, which refults from the late measurement between Greenwich and Paris, too fmall by about the fame quantity; the difference of longitude between those two places will come out 9' 19" 7:-but we cannot help withing that Mr. Dalby, while he was about it, had deduced the longitude between Greenwich and Paris from the hypothefis that the terrestrial meridians are ellipfes, of that fpecies which refults from the measurements of M. M. Condamine and Maupertuis; because we fhould then have been able to eftimate, in fome tolerable degree, the magnitude of the errors which may poffibly arife in the difference of longitude, from our uncertainty with refpect to the real figure and magnitude of the earth for it was not probable, from the reafon for which M. Bouguer's fpheroid was chofen, that any very great difference could arife between the deductions derived from it, and those derived from a fpheroid of the kind that Mr. Dalby ufed, as both were accommodated to the measurements which have been made in the middle latitudes. It was not, indeed, very likely that any confiderable variation could happen in the difference of longitude in time, whatever hypothefis was adopted. There is, therefore, nothing remarkable in the circumftance of General Roy's determination falling on one fide, and Mr. Dalby's on

the

the other fide, of Dr. Mafkelyne's deduction from aftronomical obfervations: but those who could propofe to correct, or thofe who could fuppofe that the latitude, as deduced from aftronomical obfervations, could be corrected, by terreftrial measurements, must know very little indeed of the nature either of one or the other.

On the Method of determining, from the real Probabilities of Life, the Values of contingent Reverfions, in which three Lives are involved in the Survivorship. By Mr. William Morgan, F. R. S.

Mr. Morgan tells us that the approbation which his two former papers on this fubject received from the Royal Society *, has encouraged him to purfue the doctrine of furvivorships ftill farther; and has rendered it, in fome measure, his duty to fubmit the refult of his labours to their confideration.

The problems confidered in this paper are as follows:

Prob. 1. To determine the [prefent] value of a given fum, payable if a certain life be the first that fails of three given lives. Prob. II. To determine the [prefent] value of a given fum, payable if a certain life be the fecond that fails of three given lives.

Prob. III. To determine the [prefent] value of a given fum, payable on the extinction of a certain life, if it be the last that fails of three given lives.

Prob. IV. To determine the [prefent] value of a given fum, payable on the extinction of two certain lives, fhould they be the first that fail of three given lives.

Prob. v. To find the [prefent] value of a given fum, payable on the extinction of a certain life, fhould it be the first or fecond that fails of three given lives.

Prob. vi. To find the [prefent] value of a given fum, payable on the extinction of a certain life, fhould it be the fecond or third that fails of three given lives.

Prob. VII. To find the [prefent] value of a given fum, payable on the extinction of a certain life, fhould it be the firft or laft that fails of three given lives.

Prob. VIII. To find the [prefent] value of a given fum, payable on the extinction of either of two certain lives, if either of thofe lives be the firft that fail of three given lives.

Prob. Ix. To find the [prefent] value of a given fum, payable on the extinction of either of two certain lives, fhould either of thefe lives be the fecond that fails of three given lives.

The Society honoured Mr. Morgan with Sir Godfrey Copley's gold medal for those two papers on this fubject. 6

Prob.