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means of occultations under circumstances in which (and they are the greater part) the variation in the curvature is not perceptible.

§ 8. With respect to the accuracy of observations of this nature, I have already said that the immer. sions behind the dark limb are not subject to the least error. Those on the illuminated side may be liable to some slight uncertainty if the star be not of the first or second magnitude, or if the power of the telescope be too small. The emersions from the dark limb are in general to be depended upon : whilst those from the illuminated side are the most doubtful of all*. But, even supposing that the error, in the last-mentioned case, may amount to 8 or even 10 seconds of time, will it be sufficient to conceal the variation of curvature altogether in the circumstances I have contemplated, and where the difference that it will produce may be ten or twenty times greater than the presumed


* It appears that, in some occultations, the immersions and emersions take place wholly on the dark side of the moon. See an account of the occultation of ß Virginis observed by Mr. Troughton, on May 22, 1801. Connaissance des Tems, Année xiii. page 342. Sometimes an occultation may be observed when the moon itself is not visible: as in the occultation of Venus on May 13, 1801, between 8 and 9 o'clock in the morning; the moon being then only a few hours old. Ibid. page 417. A favourable occultation may also occur during a total eclipse of the moon, when the immersions and emersions of stars of the 6th or 7th magnitude may be distinctly observed. Ibid. Année ix. page 335. B.

error? Can we, indeed, expect to obtain more certain observations* ? since a single observation of this kind accurately made in the place, for which the variation of curvature is sought, is sufficient to discover it by means of the comparison of the calculated with the observed moment.

§ 9. I shall now point out what are, in fact, the circumstances in which the differences of the parallax between two latitudes may be shown in an undoubted manner: and, in order virtually to embrace all the cases, I shall take mean quantities in the elements which enter into this investigation. Let us therefore suppose, first, that the moon's apparent semidiameter is 15'. 45′′, its equatorial parallax 57′. 40′′, and its horary motion 32′. 56′′,5 : secondly, that the occultation is observed in N. Latitude 60°, in which parallel there are three celebrated observatories, viz. Petersburg, Stockholm and Upsal: lastly, that the apparent height of the moon is 10°. If there be no variation in the curvature of the earth's surface the parallax of height will be 56'. 47", 4†: but, if the polar compression amount to of the earth's radius, the same


* I find it difficult here to give a faithful translation of the author's words: the original runs thus, "Senza che ci possiamo attenere alle "osservazioni più sicure: ed una sola fase, &c. &c." B.

+ Let p = the horizontal parallax of the moon; and h = the height of the moon: then = p. cos h the parallax of height. B.

parallax will be 56'. 38", 9*. The parallax would therefore under these circumstances experience an alteration of 8′′,5; a quantity certainly too small to be verified with accuracy by means of observations of the height of the moon. But, this slight alteration produces, in certain cases, effects that are very visible: a fact which escaped the observation of Maupertuis; who, it is true, speaks of occultations as a mean of discovering the flattening of the earth. But, he speaks of them generally; and so slightly as to class them with appulses, as being equally fit to determine it. Now, as appulses are far from being observable with such certainty, in regard to time, as the instantaneous disappearance and reappearance of stars in occultations, it is evident that Maupertuis can never have had in view the particular cases which I am about to point out, and which differ materially from ordinary ones.


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* Let a = the polar compression of the earth, supposed λ = the latitude of the place; and the radius of the equator equal to unity: then we shall have the length of any other terrestrial radius = (1—a. sin 2 a) nearly; which, being multiplied by «, will give w = (1 —a sin2 2)= the parallax of height on the supposition that the earth is an oblate spheroid. Whence we also the length of the terestrial radius at any given latitude. denote any given increase of the parallax, we shall have corresponding increase in the length of the Earth's radius.

+ Préface au discours sur la parallaxe de la lune.



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§ 10. Let us suppose AVB a portion of the circumference of the moon's disc opposite to the ob


server; C the moon's centre; and that the radius CV, which divides the arc AVB into two equal parts, coincides with the vertical circle of the place of the observer. Let us further suppose that the line MV (or the ver-sine of the arc) is equal to 60"; and that, during the occultation, the chord AB of the moon is that which the star would apparently describe if the earth were spherical; or the chord DE that which it would apparently describe if the axis of the earth were compressed. Then ML will be equal to 8′′,5 as before mentioned: and, computing from the mean rate of the horary motion of the

moon, it will be found that the duration of the occultation behind AB will be 20'. 7" of time; and, behind DE, 18'. 41"*. We therefore see how considerable is this difference of 1'.26" of time; and how well such observations are adapted not only to convince those who doubt the reality of the compression of the earth's axis, but likewise to show with very great approximation to truth the relative length of the earth's radii in different latitudes. For, in the case just mentioned, the variation of a twenty-thousandth part of that length† will cause a difference of one second of time in the duration of the occultation.

§ 11. Moreover, it is evident that the effect, of which I have been speaking, may be much greater.


*For, joining C,A, and C,B, we shall have CV-CA=CB= 15'.45", and CM (15.45"-60")=14.45"; consequently AM=MB= √(AC-CM). (AC+CM)=5.31",4; which, being multiplied by 60 in order to reduce this distance into time, will give 10.3′′, 6 for the time of the star's passing from A to M: there, fore the time of the star's passing from A to B will be equal to 2x (10.3′′,6) = 20. 7",2. But, if the depression of the earth's axis cause a variation in the apparent height of the moon equal to 8", 5, then will CL=14'. 53′′,5, consequently (as CD=AC) DL= LE=(AC-CL). (AC+CL)=5.7",7; which, being also multitiplied by will give 9', 20",5 for the time of the star's passing from D to L; therefore the time of the star's passing from D to E will be equal to 2x (9'. 20",5)=18'. 41". B.

60 32.56',5'

+Or, about 1000 feet. The mean radius of the earth being 20,898,240 English feet. B.

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