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146 Outlines of Astronomy, § 656.
147 Laplace, Système du Monde.
148 If n, n′ are the mean motions of the two planets, the expression for the disturbing force contains terms of the type
where p, p′ are integers, and the coefficient is of the order p⁓p′ in the eccentricities and inclinations. If now p and p′ are such that np⁓n′p′ is small, the corresponding inequality has a period 2π∕(np⁓n′p′), and though its coefficient is of order p⁓p′, it has the small factor np⁓np′ (or its square) in the denominator and may therefore be considerable. In the case of Jupiter and Saturn, for example, n = 109,257 in seconds of arc per annum, n′ = 43,996; 5n′ - 2n = 1,466; there is therefore an inequality of the third order, with a period (in years) = 360°∕1,466″ = 900.
149 This statement requires some qualification when perturbations are taken into account. But the point is not very important, and is too technical to be discussed.
150 ∑e2m√a = c, ∑tan2im√a = c′, where m is the mass of any planet, a, e, i are the semi-major axis, eccentricity, and inclination of the orbit. The equation is true as far as squares of small quantities, and therefore it is indifferent whether or not tan i is replaced as in the text by i.
151 Nearly the whole of the “eccentricity fund” and of the “inclination fund” of the solar system is shared between Jupiter and Saturn. If Jupiter were to absorb the whole of each fund, the eccentricity of its orbit would only be increased by about 25 per cent., and the inclination to the ecliptic would not be doubled.
152 Of tables based on Laplace’s work and published up to the time of his death, the chief solar ones were those of von Zach (1804) and Delambre (1806); and the chief planetary ones were those of Lalande (1771), of Lindenau for Venus, Mars, and Mercury (1810-13), and of Bouvard for Jupiter, Saturn, and Uranus (1808 and 1821).
153, The motion of the satellites of Uranus (chapter XII., § 253 255) is in the opposite direction. When Laplace first published his theory their motion was doubtful, and he does not appear to have thought it worth while to notice the exception in later editions of his book.
154 This statement again has to be modified in consequence of the discoveries, beginning on January 1st, 1801, of the minor planets (chapter XIII., § 294), many of which have orbits that are far more eccentric than those of the other planets and are inclined to the ecliptic at considerable angles.
155 Système du Monde, Book V., chapter VI.
156 In his paper of 1817 Herschel gives the number as 863, but a reference to the original paper of 1785 shews that this must be a printer’s error.
157 The motion of Castor has become slower since Herschel’s time, and the present estimate of the period is about 1,000 years, but it is by no means certain.
158 More precisely, counting motions in right ascension and in declination separately, he had 27 observed motions to deal with (one of the stars having no motion in declination); 22 agreed in sign with those which would result from the assumed motion of the sun.
159 The method was published by Legendre in 1806 and by Gauss in 1809, but it was invented and used by the latter more than 20 years earlier.
160 The figure has to be enormously exaggerated, the angle SσE as shewn there being about 10°, and therefore about 100,000 times too great.
161 Sir R. S. Ball and the late Professor Pritchard (§ 279) have obtained respectively ·47″ and ·43″; the mean of these, ·45″, may be provisionally accepted as not very far from the truth.
162 An average star of the 14th magnitude is 10,000 times fainter than one of the 4th magnitude, which again is about 150 times less bright than Sirius. See § 316.
163 Newcomb’s velocity of light and Nyrén’s constant of aberration (20″·4921) give 8″·794; Struve’s constant of aberration (20″·445), Loewy’s (20″·447), and Hall’s (20″·454) each give 8″·81.
164 Fundamenta Nova Investigationis Orbitae Verae quam Luna perlustrat.
165 Darlegung der theoretischen Berechnung der in den Mondtafeln angewandten Störungen.
166 E.g. in Grant’s History of Physical Astronomy, Herschel’s Outlines of Astronomy, Miss Clerke’s History of Astronomy in the Nineteenth Century, and the memoir by Dr. Glaisher prefixed to the first volume of Adams’s Collected Papers.
167 This had been suggested as a possibility by several earlier writers.
168 The discovery of a terrestrial substance with this line in its spectrum has been announced while this book has been passing through the press.
169 Observations made on Mont Blanc under the direction of M. Janssen in 1897 indicate a slightly larger number than Dr. Langley’s.
170 Catalogus novus stellarum duplicium, Stellarum duplicium et multiplicium mensurae micrometricae, and Stellarum fixarum imprimis duplicium et multiplicium positiones mediae pro epocha 1830.
171 I.e. 2·512... is chosen as being the number the logarithm of which is ·4, so that (2·512...)5∕2 = 10.
172 If L be the ratio of the light received from a star to that received from a standard first magnitude star, such as Aldebaran or Altair, then its magnitude m is given by the formula
L = (1∕2·512)m - 1 = (1∕100)(m - 1)∕5, whence m - 1 = -5∕2log L.
A star brighter than Aldebaran has a magnitude less than 1, while the magnitude of Sirius, which is about nine times as bright as Aldebaran, is a negative quantity,-1·4, according to the Harvard photometry.
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