# NUTATION

, in Astronomy, a kind of libratory motion of the earth's axis; by which its inclination to the plane of the ecliptic is continually varying, by a certain number of seconds, backwards and forwards. The whole extent of this change in the inclination of the earth's axis, or, which is the same thing, in the apparent declination of the stars, is about 19″, and the period of that change is little more than 9 years, or the space of time from its setting out from any point and returning to the same point again, about 18 years and 7 months, being the same as the period of the moon's motions, upon which it chiefly depends; being indeed the joint effect of the inequalities of the action of the sun and moon upon the spheroidal figure of the earth, by which its axis is made to revolve with a conical motion, so that the extremity of it describes a small circle, or rather an ellipse, of 19.1 seconds diameter, and 14″.2 conjugate, each revolution being made in the space of 18 years 7 months, according to the revolution of the moon's nodes.

This is a natural consequence of the Newtonian system of universal attraction; the first principle of which is, that all bodies mutually attract each other in the direct ratio of their masses, and in the inverse ratio of the squares of their distances. From this mutual attraction, combined with motion in a right line, Newton deduces the figure of the orbits of the planets, and particularly that of the earth. If this orbit were a circle, and if the earth's form were that of a perfect sphere, the attraction of the sun would have no other effect than to keep the earth in its orbit, without causing any irregularity in the position of its axis. But neither is the earth's orbit a circle, nor its body a sphere; for the earth is sensibly protuberant towards the equator, and its orbit is an ellipsis, which has the sun in its focus. Now when the position of the earth is such, that the plane of the equator passes through the centre of the sun, the attractive power of the sun acts only so as to draw the earth towards it, still parallel to itself, and without changing the position of its axis; a circumstance which happens only at the time of the equinoxes. In proportion as the earth recedes from those points, the sun also goes out of the plane of the equator, and approaches that of the one or other of the tropics; the semidiameter of the earth, then exposed to the sun, being unequal to what it was in the former case, the equator is more powerfully attracted than the rest of the globe, which causes some alteration in its position, and its inclination to the plane of the ecliptic: and as that part of the orbit, which is comprised between the autumnal and vernal equinox, is less than that which is comprised between the vernal and autumnal, it follows, that the irregularity caused by the sun, during his passage through the northern signs, is not entirely compensated by that which he causes during his passage through the southern signs; and that the parallelism of the terrestrial axis, and its inclination to the ecliptic, is thence a little altered.

The like effect which the sun produces upon the earth, by his attraction, is also produced by the moon, which acts with greater force, in proportion as she is more distant from the equator. Now, at the time when her nodes agree with the equinoxial points, her greatest latitude is added to the greatest obliquity of the ecliptic. At this time therefore, the power which causes the irregularity in the position of the terrestrial axis, acts with the greatest force; and the revolution of the nodes of the moon being performed in 18 years 7 months, hence it happens that in this time the nodes will twice agree with the equinoxial points; and consequently, twice in that period, or once every 9 years, the earth's axis will be more influenced than at any other time.

That the moon has also a like motion, is shewn by Newton, in the first book of the Principia; but he observes indeed that this motion must be very small, and scarcely sensible.

As to the history of the Nutation, it seems there have been hints and suspicions of the existence of such a circumstance, ever since Newton's discovery of the system of the universal and mutual attraction of matter; some traces of which are found in his Principia, as above mentioned.

We find too, that Flamsteed had hoped, about the year 1690, by means of the stars near his zenith, to determine the quantity of the Nutation which ought to follow from the theory of Newton; but he gave up that project, because, says he, if this effect exists, it must remain insensible till we have instruments much longer than 7 feet, and more solid and better fixed than mine. Hist. Cælest. vol. 3, pa. 113.

And Horrebow gives the following passage, extracted from the manuscripts of his master Roemer, who died in 1710, whose observations he published in 1753, un- | der the title of Basis Astronomiæ. By this paragraph it appears that Roemer suspected also a Nutation in the earth's axis, and had some hopes to give the theory of it: it runs thus; “Sed de altitudinibus non perinde certus reddebar, tàm ob refractionum varietatem quàm ob aliam nondum liquido perspectam causam; scilicet per hos duos annos, quemadmodum & alias, expertus sum esse quandam in declinationibus varietatem, quæ nec resractionibus nec parallaxibus tribui potest, sine dubio ad vacillationem aliquam poli terrestris referendam, cujus me verisimilem dare posse theoriam, observationibus munitam, spero.” Basis Astronomiæ, 1735, pa. 66.

These ideas of a Nutation would naturally present them selves to those who might perceive certain changes in the declinations of the stars; and we have seen that the first suspicions of Bradley in 1727, were that there was some Nutation of the earth's axis which caused the star g Draconis to appear at times more or less near the pole; but farther observations obliged him to search another cause for the annual variations (art. ABERRATION): it was not till some years after that he discovered the second motion which we now treat of, properly called the Nutation. See the art. Star, pa. 500 &c, where Bradley's discovery of it is given at length; to which may be farther added the following summary.

For the better explaining the discovery of the Nutation by Bradley, we must recur to the time when he observed the stars in discovering the aberration. He perceived in 1728, that the annual change of declination in the stars near the equinoxial colure, was greater than what ought to result from the annual precession of the equinoxes being supposed 50″, and calculated in the usual way; the star h Ursæ Majoris was in the month of September 1728, 20″ more south than the preceding year, which ought to have been only 18″; from whence it would follow that the precession of the equinoxes should be 55″1/2 instead of 50″, without ascribing the difference between the 18 and 20″ to the instrument, because the stars about the solstitial colure did not give a like difference. Philos. Trans. vol. 35, pa. 659.

In general, the stars situated near the equinoctial colure had changed their declination about 2″ more than they ought by the mean precession of the equinoxes, the quantity of which is very well known, and the stars near the solstitial colure the same quantity less than they ought; but, Bradley adds, whether these small variations arise from some regular cause, or are occasioned by some change in the sector, I am not yet able to determine. Bradley therefore ardently continued his observations for determining the period and the law of these variations; for which purpose he resided almost continually at Wansted till 1732, when he was obliged to repair to Oxford to succeed Dr. Halley; he still continued to observe with the same exactness all the circumstances of the changes of declination in a great number of stars. Each year he saw the periods of the aberration confirmed according to the rules he had lately discovered; but from year to year he found also other differences; the stars situated between the vernal equinox and the winter solstice approached nearer to the north pole, while the opposite ones receded farther from it: he began therefore to suspect that the action of the moon upon the elevated equatorial parts of the earth might cause a variation or libration in the earth's axis: his sector having been left fixed at Wansted, he often went there to make observations for many years, till the year 1747, when he was fully satisfied of the cause and effects, and account of which he then communicated to the world. Philos. Trans. vol. 45, an. 1748.

“On account of the inclination of the moon's orbit to the ecliptic, says Dr. Maskelyne (Astronomical Observations 1776, pa. 2), and the revolution of the nodes in antecedentia, which is performed in 18 years and 7 months, the part of the precession of the equinoxes, owing to her action, is not uniform: but subject to an equation, whose maximum is 18″: and the obliquity of the ecliptic is also subject to a periodical equation of 9″.55; being greater by 19.1″ when the moon's ascending node is in Aries, than when it is in Libra. Both these effects are represented together, by supposing the pole of the earth to describe the periphery of an ellipsis, in a retrograde manner, during each period of the moon's nodes, the greater axis, lying in the solstitial colure, being 19.1″, and the lesser axis, lying in the equinoctial colure, 14.2″; being to the greater, as the cosine of double the obliquity of the ecliptic to the cosine of the obliquity itself. This motion of the pole of the earth is called the Nutation of the earth's axis, and was discovered by Dr. Bradley, by a series of observations of several stars made in the course of 20 years, from 1727 to 1747, being a continuation of those by which he had discovered the aberration of light. But the exact law of the motion of the earth's axis has been settled by the learned mathematicians d'Alembert, Euler, and Simpson, from the principles of gravity. The equation hence arising in the place of a fixed star, whether in longitude, right-ascension, or declination (for the latitudes are not affected by it) has been sometimes called Nutation, and sometimes Deviation.” And again (says the Doctor, pa. 8), the above “quantity 19.1″, of the greatest Nutation of the earth's axis in the solstitial colure, is what I found from a scrupulous calculation of all Dr. Bradley's observations of g Draconis, which he was pleased to communicate to me for that purpose. From a like examination of his observation of h Ursæ majoris, I found the lesser axis of the ellipsis of Nutation to be 14.1″, or only (1/10)th of a second less than what it should be from the observations of g Draconis. But the result from the observations of g Draconis is most to be depended upon.”

Mr. Machin, secretary of the Royal Society, to whom Bradley communicated his conjectures, soon perceived that it would be sufficient to explain, both the Nutation and the change of the precession, to suppose that the pole of the earth described a small circle. He stated the diameter of this circle at 18″, and he supposed that it was described by the pole in the space of one revolution of the moon's nodes. But later calculations and theory, have shewn that the pole describes a small ellipsis, whose axes are 19.1″ and 14.2″, as above mentioned.

To shew the agreement between the theory and observations, Bradley gives a great multitude of observations of a number of stars, taken in different positions; and out of more than 300 observations which he made, he found but 11 which were different from the mean by | so much as 2″. And by the supposition of the elliptic rotation, the agreement of the theory with observation comes out still nearer.

By the observations of 1740 and 1741, the star h Ursæ majoris appeared to be 3″ farther from the pole than it ought to be according to the observations of other years. Bradley thought this difference arose from some particular cause; which however was chiefly the fault of the circular hypothesis. He suspected also that the situation of the apogee of the moon might have some influence on the Nutation. He invited therefore the mathematicians to calculate all these effects of attraction, which has been ably done by d'Alembert, Euler, Walmesley, Simpson, and others; and the astronomers to continue to observe the positions of the smallest stars, as well as the largest, to discover the physical derangements which they may suffer, and which had been observed in some of them.

Several effects arise from the Nutation. The first of these, and that which is the most easily perceived, is the change in the obliquity of the ecliptic; the quantity of which ought to be varied from that cause by 18″ in about 9 years. Accordingly, the obliquity of the ecliptic was observed in 1764 to be 23° 28′ 15″, and in 1755 only 23° 28′ 5″: not only therefore had it not diminished by 8″, as it ought to have done according to the regular mean diminution of that obliquity; but it had even augmented by 10″; making together 18″, for the effect of the Nutation in the 9 years.

The Nutation changes equally the longitudes, the right-ascensions, and the declinations of the stars, as before observed; it is the latitudes only which it does not affect, because the ecliptic is immoveable in the theory of the Nutation.

Dr. Bradley illustrates the foregoing theory of Nutation in the following manner. Let P represent the mean place of the pole of the equator, about which point, as a centre, suppose the true pole to move in the small circle ABCD, whose diameter is 18″. Let E be the pole of the ecliptic, and EP be equal to the mean distance between the poles of the equator and ecliptic; and suppose the true pole of the equator to be at A, when the moon's ascending node is in the beginning of Aries; and at B, when the node gets back to Capricorn; and at C, when the same node is in Libra: at which time the north pole of the equator being nearer the north pole of the ecliptic, by the whole diameter of the little circle AC, equal to 18″; the obliquity of the ecliptic will then be so much less than it was, when the moon's ascending node was in Aries. The point P is supposed to move round E, with an equal retrograde motion, answerable to the mean precession arising from the joint actions of the sun and moon: while the true pole of the equator moves round P, in the circumference ABCD, with a retrograde motion likewise, in a period of the moon's nodes, or of 18 years and 7 months. By this means, when the moon's ascending node is in Aries, and the true pole of the equator, at A, is moving from A towards B; it will approach the stars that come to the meridian with the sun about the vernal equinox, and recede from those that come with the sun near the autumnal equinox, faster than the mean pole P does. So that, while the moon's node goes back from Aries to Capricorn, the apparent precession will seem so much greater than the mean, as to cause the stars that lie in the equinoctial colure to have altered their declination 9″, in about 4 years and 8 months, more than the mean precession would do; and in the same time, the north pole of the equator will seem to have approached the stars that come to the meridian with the sun of our winter solstice about 9″, and to have receded as much from those that come with the sun at the summer solstice.

Thus the phenomena before recited are in general conformable to this hypothesis. But to be more particular; let S be the place of a star, PS the circle of declination passing through it, representing its distance from the mean pole, and <*> PS its mean right-ascension. Thus if O and R be the points where the circle of declination cuts the little circle ABCD, the true pole will be nearest that star at O, and farthest from it at R; the whole difference amounting to 18″, or to the diameter of the little circle. As the true pole of the equator is supposed to be at A, when the moon's ascending node is in Aries; and at B, when that node gets back to Capricorn; and the angular motion of the true pole about P, is likewise supposed equal to that of the moon's node about E, or the pole of the ecliptic; since in these cases the true pole of the equator is 90 degrees before the moon's ascending node, it must be so in all others.

When the true pole is at A, it will be at the same distance from the stars that lie in the equinoctial colure, as the mean pole P is; and as the true pole recedes back from A towards B, it will approach the stars which lie in that part of the colure represented by P<*>, and recede from those that lie in P; not indeed with an equable motion, but in the ratio of the sine of the distance of the moon's node from the beginning of Aries. For if the node be supposed to have gone backwards from Aries 30°, or to the beginning of Pisces, the point which represents the place of the true pole will, in the mean time, have moved in the little circle through an arc, as AO, of 30° likewise; and would therefore in effect have approached the stars that lie in the equinoctial colure P<*>, and have receded from those that lie in P by 4 1/<*> seconds, which is the sine of 30° to the radius AP. For if a perpendicular fall from O upon AP, it may be conceived as part of a great circle, passing through the true pole and any star lying in the equinoctial colure. Now the same proportion that holds in these stars, will obtain likewise in all others; and from hence we may collect a general rule for finding how much nearer, or farther, any star is to, or from, the mean pole, in any given position of the moon's node.

For, If from the right-ascension of the star, we subtract the distance of the moon's ascending node from Aries; then radius will be to the sine of the remainder, as 9″ is to the number of seconds that the star is nearer to, or farther from, the true, than the mean pole.

This motion of the true pole, about the mean at P, will also produce a change in the right-ascension of the | stars, and in the places of the equinoctial points, as well as in the obliquity of the ecliptic; and the quantity of the equations, in either of these cases, may be easily computed for any given position of the moon's nodes.

Dr. Bradley then proceeds to find the exact quantity of the mean precession of the equinoctial points, by comparing his own observations made at Greenwich, with those of Tycho Brahe and others; the mean of all which he states at 1 degree in 71 1/2 years, or 50 1/3″ per year; in order to shew the agreement of the foregoing hypothesis with the phenomena themselves, of the alterations in the polar distances of the stars; the conclusions from which approach as near to a coincidence as could be expected on the foregoing circular hypothesis, the diameter of which is 18″; instead of the more accurate quantity 19.1″, as deduced by Dr. Maskelyne, and the elliptic theory as determined by the mathematicians, in which the greater axis (19.1″) is to the less axis (14.2″), as the cosine of the greatest declination is to the cosine of double the same.

To give an idea now of the Nutation of the stars, in longitude, right-ascension, and declination; suppose the pole of the equator to be at any time in the point O, also S the place of any star, and OH perpendicular to AE: then, like as AE is the solstitial colure when the pole of the equator was at A, and the longitude of the star S equal to the angle AES; so OE is the solstitial colure when that pole is at O, and the longitude is then only the angle OES; less than before by the angle AEO, which therefore is the Nutation in longitude: counting the longitudes from the solstitial instead of the equinoctial colure, from which they differ equally by 90 degrees, and therefore have the same difference AEO. Now the angle AEO will be as the line HO = sin. AO to radius PB = sin. AO X PB = sin. AO X 9″; therefore as , since AO is equal to longitude of the moon's node. This expression therefore gives the Nutation in longitude, supposing the maximum of Nutation, with Bradley, to be 18″; and it is negative, or must be subtracted from the mean longitude of the stars, when the moon's node is in the first 6 signs of its longitude, but additive in the latter 6, to give the true apparent longitude.

This equation of the Nutation in longitude is the same for all the stars; but that for the declination and right ascension is various for the different stars. In the foregoing figure, PS is the mean polar distance, or mean codeclination, of the star S, when the true place of the pole is O; and SO the apparent codeclination; also, the angle SPE is the mean right-ascension, and SOE the apparent one, counted from the solstitial colure; consequently OPS or OPF the difference between the right-ascension of the star and that of the pole, which is equal to the longitude of the node increased by 3 signs or 90 degrees; supposing OF to be a small arc perpendicular to the circle of declination PFS; then is SF = SO, and PF the Nutation in declination, or the quantity the declination of the star has increased; but radius 1 : 9″ :: cosin. OPF : PF = 9″ X cos. OPF; so that the equation of decli- nation will be found by multiplying 9″ by the sine of the star's right-ascension diminished by the longitude of the node; for that angle is the complement of the angle SPO. This Nutation in declination is to be added to the mean declination to give the apparent, when its argument does not exceed 6 signs; and to be subtracted in the latter 6 signs. But the contrary for the stars having south declination.

To calculate the Nutation in right-ascension, we must find the difference between the angle SOE the apparent, and SPE the mean right-ascension, counted from the solstitial colure EO. Now the true rightascension SOE is equal to the difference between the two variable angles GOE and GOS; the former of which arises from the change of one of the variable circles EO, and depends only on the situation of the node or of that of the pole O; the latter GOS depends on the angle GPS which is the difference between the right-ascension of the star and the place of the pole O. Now in the spherical triangle GPE, which changes into GOE, the side GE and the angle G remain constant, and the other parts are variable; hence therefore the small variation PO of the side next the constant angle G, is to the small variation of the angle opposite to the constant side GE, as the tangent of the side PE opposite to the constant angle, is to the sine of the angle GPE opposite to the constant side; that is, as , the difference between the angles GOE and GPE. This is the change which the Nutation PO produces in the angle GPE, being the first part of the Nutation sought, and is common to all the stars and planets. It is to be subtracted from the mean right-ascension in the first 6 signs of the longitude of the node, and added in the other six.

In like manner is found the change which the Nutation produces in the other part of the right-ascension SPE, that is, in the angle SPG, which becomes SOG by the effect of the Nutation. This small variation will be calculated from the same analogy, by means of the triangle SOG, in which the angle G is constant, as well as the side SG, whilst SP changes into SO. Hence therefore, tang. SP : sin. SPG :: 9″ : variation of SPG, that is, the cotangent of the declination is to the cosine of the distance between the star and the node, as 9″ are to the quantity the angle SPG varies in becoming the angle SOG, being the second part of the Nutation in right-ascension; and if there be taken for the argument, the right-ascension of the star minus the longitude of the node, the equation will be subtractive in the first and last quadrant of the argument, and additive in the 2d and 3d, or from 3 to 9 signs. But the contrary for stars having south declination.

This second part of the Nutation in right-ascension affects the return of the sun to the meridian, and therefore it must be taken into the account in computing the equation of time. But the former part of the Nutation does not enter into that computation; because it only changes the place of the equinox, without changing the point of the equator to which a star corresponds, and consequently without altering the duration of the returns to the meridian. |

All these calculations of the Nutation, above explained, are upon Machin's hypothesis, that the pole describes a circle; however Bradley himself remarked that some of his observations differed too much from that theory, and that such observations were found to agree better with theory, by supposing that the pole, instead of the circle, describes an ellipse, having its less axis DB = 16″ in the equinoctial colure, and the greater axis AC = 18″, lying in the solstitial colure. But as even this correction was not sufficient to cause all the inequalities to disappear entirely, Dr. Bradley referred the determination of the point to theoretical and physical investigation. Accordingly several mathematicians undertook the task, and particularly d'Alembert, in his Recherches fur la précession des equinoxes, where he determines that the pole really describes an ellipse, and that narrower than the one assumed above by Bradley, the greater axis being to the less, as the cosine of 23° 28′ to the cosine of double the same. And as Dr. Maskelyne found, from a more accurate reduction of Bradley's observations, that the maximum of the Nutation gives 19.1″ for the greater axis, therefore the above proportion gives 14.2″ for the less axis of it; and according to these data, the theory and observations are now found to agree very near together.

See La Lande's Astron. vol. 3, art. 2874 &c, where he makes the corrections for the ellipse. He observes however that by the circular hypothesis alone, the computations may be performed as accurately as the observations can be made; and he concludes with some corrections and rules for computing the Nutation in the elliptic theory.

The following set of general tables very readily give the effect of Nutation on the elliptical hypothesis; they were calculated by the late M. Lambert, and are taken from the Connoissance des Temps for the year 1788.

 General Tables for Nutation in the Ellipse. Table 1. Table 2. Table 3. De- 0.6 17 2.8 De- 0.6 1.7 2.8 De- 0.6 1.7 2.8 grees + - + - + - grees + - + - + - grees - + - + - + ″ ″ ″ ″ ″ ″ ″ ″ ″ 0 0.00 3.93 6.80 30 0 0.00 0.58 1.00 30 0 0.00 7.71 13.36 30 1 0.14 4.04 6.86 29 1 0.02 0.59 1.01 29 1 0.27 7.95 13.50 29 2 0.27 4.16 6.93 28 2 0.04 0.61 1.02 28 2 0.54 8.18 13.62 28 3 0.41 4.28 6.99 27 3 0.06 0.63 1.02 27 3 0.81 8.40 13.75 27 4 0.55 4.39 7.06 26 4 0.08 0.64 1.03 26 4 1.08 8.63 13.87 26 5 0.68 4.50 7.11 25 5 0.10 0.66 1.04 25 5 1.35 8.85 13.98 25 6 0.82 4.61 7.17 24 6 0.12 0.68 1.05 24 6 1.61 9.07 14.10 24 7 0.95 4.72 7.23 23 7 0.14 0.69 1.06 23 7 1.88 9.29 14.20 23 8 1.11 4.83 7.28 22 8 0.16 0.71 1.07 22 8 2.15 9.50 14.31 22 9 1.23 4.94 7.33 21 9 0.18 0.72 1.07 21 9 2.41 9.71 14.41 21 10 1.36 5.05 7.38 20 10 0.20 0.74 1.08 20 10 2.68 9.92 14.50 20 11 1.50 5.15 7.42 19 11 0.22 0.75 1.09 19 11 2.94 10.12 14.59 19 12 1.63 5.25 7.47 18 12 0.24 0.77 1.09 18 12 3.21 10.32 14.67 18 13 1.77 5.35 7.51 17 13 0.26 0.78 1.10 17 13 3.47 10.52 14.76 17 14 1.90 5.45 7.55 16 14 0.28 0.80 1.11 16 14 3.73 10.72 14.83 16 15 2.03 5.55 7.58 15 15 0.30 0.81 1.11 15 15 3.99 10.91 14.90 15 16 2.16 5.65 7.62 14 16 0.32 0.83 1.12 14 16 4.25 11.10 14.97 14 17 2.30 5.74 7.65 13 17 0.34 0.84 1.12 13 17 4.51 11.28 15.03 13 18 2.43 5.83 7.68 12 18 0.35 0.85 1.13 12 18 4.77 11.47 15.09 12 19 2.56 5.92 7.71 11 19 0.37 0.87 1.13 11 19 5.02 11.65 15.15 11 20 2.68 6.01 7.73 10 20 0.39 0.88 1.13 10 20 5.28 11.82 15.20 10 21 2.81 6.10 7.75 9 21 0.41 0.89 1.14 9 21 5.53 11.99 15.24 9 22 2.94 6.19 7.76 8 22 0.43 0.91 1.14 8 22 5.78 12.16 15.28 8 23 3.07 6.27 7.77 7 23 0.45 0.92 1.14 7 23 6.03 12.32 15.32 7 24 3.19 6.35 7.79 6 24 0.47 0.93 1.14 6 24 6.28 12.48 15.35 6 25 3.32 6.43 7.80 5 25 0.49 0.94 1.15 5 25 6.52 12.64 15.37 5 26 3.44 6.51 7.82 4 26 0.50 0.95 1.15 4 26 6.76 12.79 15.39 4 27 3.56 6.58 7.83 3 27 0.52 0.96 1.15 3 27 7.01 12.94 15.41 3 28 3.69 6.66 7.84 2 28 0.54 0.97 1.15 2 28 7.25 13.09 15.42 2 29 3.81 6.73 7.85 1 29 0.56 0.99 1.15 1 29 7.48 13.23 15.43 1 30 3.93 6.80 7.85 0 30 0.58 1.00 1.15 0 30 7.71 13.36 15.43 0 + - + - + - De- + - + - + - De- - + - + - + De- 5.11 4.10 3.9 grees 5.11 4.10 3.9 grees 5.11 4.10 3.9 grees
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The Use of the Tables.

The right-ascension of a star minus the moon's mean longitude, gives the argument of the first of these three tables. The sum of the same two quantities gives the argument of the 2d table. Then the sum or the difference of the quantities found with these two arguments, will give the correction to be applied to the mean declination of the star, if it is north declination; but if it is southern, the signs + or - are to be changed into - and +.

From each of those two arguments for the declination subtracting 3 signs, or 90°, gives the arguments for correcting the right-ascension; the sum or difference of the quantities found, with these two arguments, in tables 1 and 2, is to be multiplied by the tangent of the star's declination, and to the product is to be added the quantity taken out of table 3, the argument of which is the mean longitude of the moon's ascending node: when the declination of the star is south, the tangent will be negative.

Example. To find the Nutation in right-ascension and declination for the star a Aquilæ, the 1st of July 1788.

 Right-ascension of the star 9s 25° 7′ Long. of the moon's node 8 15 40 ″ Diff. being argument 1, 1 9 27 + 4.99 Sum, argument 2, 6 10 47 - 0.22 Correction of the declination + 4.77
The above two arguments being each diminished by 3 signs, give,
 s ° ′  ″ Argument 1 10 9 27 - 6.06 Argument 2 3 10 47 + 1.13 - 4.93 Declin. of star north, its tangent 0.146 The product is - 0.72 Long. of the 's node, argum. 3 + 14.94 Correction of right-ascension + 14.22

In general, let denote the longitude of the moon's ascending node; r the right-ascension of a star or planet; d its declination; the Nutation in declination and rightascension will be expressed by the two following formulæ; viz, the Nutation in declination = 7″.85 X sin. (r - ) + 1″.15 X sin. (r + ); and the Nutation in right-ascension = [7″.85 X sin. (r - - 90°) + 1″.15 X sin. (r + - 90°)] X tang. d - 15″.43 X sin. .

For the mathematical investigation of the effects of universal attraction, in producing the Nutation, &c, see d'Alembert's Recherches fur la Precession des Equinoxes; Silvabelle's Treatise on the Precession of the Equinoxes &c, in the Philos. Trans. an. 1754, p. 385; Walmesley's treatise De Præcessione Equinoctiorum et Axis Terræ Nutatione, in the Philos. Trans. an. 1756, pa. 700; Simpson's Miscellaneous Tracts, pa. 1; and other authors.

S

STEAM. The observations on the different degrees of temperature acquired by water in boiling, under different pressures of the atmosphere, and the formation of the vapour from water under the receiver of an airpump, when, with the common temperatures, the pressure is diminished to a certain degree, have taught us that the expansive force of vapour or Steam is different in the different temperatures, and that in general it increases in a variable ratio as the temperature is raised.

But there was wanting, on this important subject, a series of exact and direct experiments, by means of which, having given the degree of temperature in boiling water, we may know the expansive force of the Steam rising from it; and vice versa. There was wanting also an analytical theorem, expressing the relation between the temperature of boiling water, and the pressure with which the force of its Steam is in equilibrium. These circumstances then have lately been accomplished by M. Betancourt, an ingenious Spanish philosopher, the particulars of which are described in a memoir communicated to the French Academy of Sciences in 1<*>90, and ordered to be printed in their collection of the Works of Strangers.

The apparatus which M. Betancourt makes use of, is a copper vessel or boiler, with its cover firmly soldered on. The cover has three holes, which close up with screws: the first is to put the water in and out; through the second passes the stem of a thermometer, which has the whole of its scale or graduations above the vessel, and its ball within, where it is immersed either in the water or the Steam according to the different circumstances; through the third hole passes a tube making a communication between the cavity of the boiler and one branch of an inverted syphon, which, containing mercury, acts as a barometer for measuring the pressure of the elastic vapour within the boiler. There is a fourth hole, in the side of the vessel, into which is inserted a tube, with a turn-cock, making a communication with the receiver of an air-pump, for extracting the air from the boiler, and to prevent its return.

The apparatus being prepared in good order, and distilled water introduced into the boiler by the first hole, and then stopped, as well as the end of the inverted syphon or barometer, M. Betancourt surrounded the boiler with ice, to lower the temperature of the water to the freezing point, and then extracting all the air from the boiler by means of the air-pump, the difference between the columns of mercury in the two branches of the barometer is the measure of the spring of the vapour arising from the water in that temperature. Then, lighting the fire below the boiler, he raised gradually the temperature of the water from 0 to 110 degrees of Reaumur's thermometer; being the same as from 32 to 212 degrees of Fahrenheit's; and for each degree of elevation in the temperature, he observed the height of the column of mercury which measured the elasticity or pressure of the vapour.

The results of M. Betancourt's experiments are con- | tained in a table of four columns, which are but little different, according to the different quantities of water in the vessel. It is here observable, that the increase in the expansive force of the vapour, is at first very slow; but gradually increasing faster and faster, till at last it becomes very rapid. Thus, the strength of the vapour, at 80 degrees, is only equal to 28 French inches of mercury; but at 110 degrees it is equal to no less than 98 inches, that is 3 times and a half more for the increase of only 30 degrees of heat.

To express analytically the relation between the degrees of temperature of the vapour, and its expansive force, this author employs a method devised by M. Prony. This method consists in conceiving the heights of the columns of mercury, measuring the expansive force, to represent the ordinates of a curve, and the degrees of heat as the abscisses of the same; making the ordinates equal to the sum of several logarithmic ones, which contain two indeterminates, and determining these quantities so that the curve may agree with a good number of observations taken throughout the whole extent of them. Then constructing the curve which results immediately from the experiments, and that given by the formula, these two curves are found to coincide almost perfectly together; the small differences being doubtless owing to the little irregularities in the experiments and in dividing the scale; so that the phenomena may be considered as truly represented by the formula.

M. Betancourt made also experiments with the vapour from spirit of wine, similar to those made with water; constructing the curve, and giving the formula proper to the same. From which is derived this remarkable result, that, for any one and the same degree of heat, the strength of the vapour of spirit of wine, is to that of water, always in the same constant ratio, viz, that of 7 to 3 very nearly; the strength of the former being always 2 1/3 times the strength of the latter, with the same degree of heat in the liquid.

Of the Formula, or Equation to the Curve.

The equation to the curve of temperature and pressure, denoting the relation between the abscisses and ordinates, or between the temperature of the vapour and its strength, is, for water, . Where x denotes the abscisses of the curve, or the degrees of Reaumur's thermometer; and y the corresponding ordinates, or the heights of the column of mercury in Paris inches, representing the strength or elasticity of the vapour answering to the number x of degrees of the thermometer. Then, by comparing this formula with a proper number of the experiments, the values of the constant quantities come out as below:

b = 10.
a = 0.068831
c = 0.019438
<*> = 0.013490
e = - 4.689760
c″ = 0.058622
e′ = - 3.937600
c‴ = 0.049220

Hence it is evident by inspection, that the terms of the equation are very easy to calculate. For, b being the radix or root of the common system of logarithms, and all the terms on the second side of the equation being the powers of b, these terms are consequently the tabular natural numbers having the variable exponents for their logarithms. Now as x rises only to the first power, and is multiplied by a constant number, and another constant number being added to the product, gives the variable exponent, or logarithm; to which then is immediately found the corresponding natural number in the table of logarithms.

In the above formula, the two last terms may be entirely omitted, as very small, as far as to the 90th degree of the thermometer; and even above that temperature those two terms make but a small part of the whole formula.

And for the spirit of wine the formula is . Where x and y, as before, denote the absciss and ordinate of the curve, or the temperature and expansive force of the vapour from the spirit of wine; also the values of the constant quantities are as below:

b = 10.
a = - 0.04853
c = 0.02393
a′ = - 0.63414
c′ = - 0.096532
e = - 2.509542
c″ = 0.046473
e′ = - 1.790192
c‴ = 0.029448
A = 1.12647

This formula is of the same nature as the former, having also the like ease and convenience of calculation; and perhaps more so; as the second term ba′ + c′x, having its exponent wholly negative, soon diminishes to no value, so as to be omitted from the 10th degree of temperature; also the difference between the last two terms - be + c″x + be′ + c‴x may be omitted till the 70th degree, for the same reason. So that, to the 10th degree of temperature the theorem is only ; and from the 10th to the 70th degree it is barely ; after which, for the last 15 or 20 degrees, for great accuracy, the last two terms may be taken in.

A compendium of the table of the experiments here follows, for the vapour of both water and spirit of wine, the temperature by Reaumur's thermometer, and the barometer in French inches. |

Table of the Temperature and Strength of the Vapour of Water and Spirit of Wine, by Reaumur's Thermometer, and French Inches.
 Height of the Barometer for Height of the Barometer for Degr. of Vapour of Vapour of Deg. of Vapour Vapour of Reau. Ther. Water. Spirit of Wine. Reau. Ther. of Water. Spirit of Wine. 1 0.0176 0.0043 56 7.6948 18.4420 2 0.0346 0.0208 57 8.1412 19.5081 3 0.0538 0.0478 58 8.6221 20.6286 4 0.0747 0.0837 59 9.1071 21.6071 5 0.1038 0.1279 60 9.6280 23.0544 6 0.1211 0.1794 61 10.1767 24.3451 7 0.1508 0.2377 62 10.7098 25.6107 8 0.1741 0.3024 63 11.3602 27.1444 9 0.2073 0.3733 64 11.9976 28.6483 10 0.2304 0.4502 65 12.6687 30.2262 11 0.2681 0.5130 66 13.3743 31.8795 12 0.3039 0.6058 67 14.1161 33.6114 13 0.3419 0.7040 68 14.8958 35.4258 14 0.3877 0.8077 69 15.7153 37.3232 15 0.4258 0.9172 70 16.577 39.3076 16 0.4778 1.0330 71 17.482 41.3807 17 0.5208 1.1553 72 18.433 43.5465 18 0.5730 1.2846 73 19.433 45.8042 19 0.6283 1.4212 74 20.485 48.1589 20 0.6872 1.5655 75 21.587 50.6096 21 0.7497 1.7180 76 22.746 53.1593 22 0.8159 1.8791 77 23.965 55.8095 23 0.8863 2.0494 78 25.260 58.3968 24 0.9610 2.2293 79 26.588 61.3057 25 1.0402 2.4194 80 28.006 64.3524 26 1.1239 2.6202 81 29.455 67.4095 27 1.2127 2.8325 82 30.980 70.4967 28 1.3068 3.0568 83 32.575 73.7647 29 1.4065 3.2937 84 34.251 77.0764 30 1.5019 3.5441 85 35.984 80.4708 31 1.6333 3.8087 86 37.800 83.9351 32 1.7413 4.0883 87 39.697 87.4625 33 1.8671 4.3837 88 41.642 91.1366 34 1.9980 4.6958 89 43.730 94.6580 35 2.1374 5.0256 90 45.870 98.2764 36 2.2846 5.3741 91 48.092 37 2.4401 5.6423 92 50.408 38 2.6045 6.1315 93 52.785 39 2.7780 6.5426 94 55.253 40 2.9711 6.9770 95 57.801 41 3.1544 7.4360 96 60.423 42 3.3583 7.9211 97 63.108 43 3.5735 8.4336 98 65.877 44 3.8005 8.9751 99 68.692 45 4.0399 9.5476 100 71.552 46 4.2922 10.1516 101 74.444 47 4.5582 10.7906 102 77.359 48 4.8386 11.4606 103 80.268 49 5.1346 12.1800 104 83.259 50 5.4453 12.9340 105 85.992 51 5.7706 13.7300 106 88.735 52 6.1194 14.5720 107 91.367 53 6.4834 15.4610 108 93.815 54 6.8667 16.4000 109 96.039 55 7.2798 17.3930 110 98.356
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M. Betancourt deduces several useful and ingenious consequences and applications from this course of experiments. He shews, for instance, that the effect of Steam engines must, in general, be greater in winter than in summer; owing to the different degrees of temperature in the water of injection. And from the very superior strength of the vapour of spirit of wine, over that of water, he argues that, by trying other fluids, some may be found, not very expensive, whose vapour may be so much stronger than that of water, with the same degree of heat, that it may be substituted instead of water in the boilers of Steam-engines, to the great saving in the very heavy expence of fuel: nay, he even declares, that spirit of wine itself might thus be employed in a machine of a particular construction, which, with the same quantity of fuel, and without any increase of expence in other things, shall produce an effect greatly superior to what is obtained from the steam of water. He makes several other observations on the working and improvement of Steamengines.

Another use of these experiments, deduced by M. Betancourt, is, to measure the height of mountains, by means of a thermometer, immersed in boiling water, which he thinks may be done with a precision equal, if not superior, to that of the barometer. As soon as I had obtained exact results of my experiments, says he, and was convinced that the degree of heat received by water depends absolutely on the pressure upon its surface, I endeavoured to compare my observations with such as have been made on mountains of different heights, to know what is the degree of heat which water can receive when the barometer stands at a determinate height; but from so few observations having been made of this kind, and the different ways employed in graduating instruments, it is difficult to draw any certain consequences from them.

The first observation which M. Betancourt compared with his experiments, is one mentioned in the Memoirs of the Academy of Sciences, anno 1740, page 92. It is there said, that M. Monnier having made water boil upon the mountain of Canigou, where the barometer stood at 20.18 inches, the thermometer immersed in this water stood at a point answering to 71 degrees of Reaumur: whereas in M. Betancourt's table of experiments, at an equal pressure upon the surface of the water, the thermometer stood at 73.7 degrees. This difference he thinks is owing partly to the want of precision in the observation, and partly to the different method of graduating the thermometer, and the neglect of purging the barometer tube of air.

M. Betancourt next compared his experiments with some observations made by M. De Luc on the tops of several mountains; in which, after reducing the scales of this gentleman to the same measures as his own, he finds a very near degree of coinoidence indeed. The following table contains a specimen of these comparisons, the instances being taken at random from De Luc's treatise on the Modifications of the Atmosphere.

 Degrees of Heat in Boiling Water upon the Heat of the Tops of Mountains, observed by De Luc. Water in M. Places of Heat of Height of Heat of the Betancourt's Observation. the air. the Bar Wa. by Th. Experim. Beaucaire 14 1/4 28.248 80.37 80.29 Geneva 12 1/2 27.056 79.33 79.33 Grange Town 16 1/4 24.510 77.11 77.42 Lans le Bourg 24-145 77.18 77.14 Grange le F. 15 24.089 76.76 77.09 Grenairon 10 1/4 20.427 73.26 73.89 Glaciere de B. 6 1/2 19.677 72.56 73.24
Where it is remarkable, that the difference between the two is of no consequence in such matters.

Many other advantages might be deduced from the exact knowledge of the effect which the pressure of the atmosphere has upon the heat which water can receive; one of which, M. Betancourt observes, is of too great importance in physics not to be mentioned. As soon as the thermometer became known to philosophers, almost every one endeavoured to find out two fixed points to direct them in dividing the scale of the instrument; having found that those of the freezing and boiling of water were nearly constant in different places, they gave these the preference over all others: but having discovered that water is capable of receiving a greater or less quantity of heat, according to the pressure of the atmosphere upon its surface, they felt the necessity of fixing a certain constant value to that pressure, which it was almost generally agreed should be equal to a column of 28 French inches of mercury. This agreement however did not remove all the difficulties. For instance, if it were required to construct at Madrid a thermometer that might be comparable with another made at Paris, the thing would be found impossible by the means hitherto known, because the barometer never rises so high as 27 inches at Madrid; and it was not certainly known how much the scale of the thermometer ought to be increased to have the point of boiling water in a place where the barometer is at 28 inches. But by making use of the foregoing observations, the thing appears very easy, and it is to be hoped that by the general knowledge of them, thermometers may be brought to great perfection, the accurate use of which is of the greatest importance in physics.

Besides, without being confined to the height of the barometer in the open air, in a given place, we may regulate a thermometer according to any one assigned heat of water, by means of such an apparatus as M. Betancourt's. For, in order to graduate a thermometer, having a barometer ready divided; it is evident that by knowing, from the foregoing table of experiments, the degree of heat answering to any one expansive force, we can thence assign the degree of the thermometer corresponding to a certain height of the barometer. A determination admitting of great precision, especially in the higher temperatures, where the motion of the barometer is so considerable in respect to that of the thermometer.

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Entry taken from A Mathematical and Philosophical Dictionary, by Charles Hutton, 1796.

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