There is still another method which has never yet been employed, so far as we know, though it seems to offer decided advantages. It consists in finding the general term in the development of this important function by means of Laplace's Equation, technically so called. Having obtained the expansion of the general term to the order required, all the other terms would, of course, follow immediately from it.* The numerical calculations can be performed by any ordinary computer, since the formula are all of an ordinary algebraic character. Since the dynamical stability of the planetary system is dependent on the character of the secular inequalities, it is necessary to give a brief account of the labors of several distinguished geometers who have devoted their attention to this class of perturbations. Lagrange and Laplace were the master minds in this department of physical astronomy. The integration of the differential equations to which they arrived was extremely difficult, and, for a time, it arrested the progress of both geometers. The genius of Lagrange—whose command of analysis was perhaps never equalled by any other mathema. tician—was equal to the task, however, for he succeeded in transforming the variables so as to reduce the differential equations to the linear form. The perihelia and eccentricities are usually treated together, and also the nodes and inclinations. Laplace applied, successfully, Lagrange's méthod of integration to his own expressions for the 'nodes and inclinations; and he also extended it to the perihelia and eccentricities. Laplace finally succeeded in demonstrating the following elegant and important theorems: “If the mass of each planet be multiplied by the square of the eccentricity, and this product by the square root of the mean distance, the sum of these quantities will always retain the same magnitude.” This sum is found to be small for any given epoch, and hence it must always remain small. “If the mass of each planet be multiplied by the square of the tangent of the orbit's - *A memoir is now preparing on this method of development, in which the terms are carried to the eighth order of small quantities, and it will probably soon be ready for publication. It inclination to a fixed plane, and this product by the square root of the mean distance, the sum of such quantities will continue invariable." This sum is found to be small for any given epoch.* The ultimate condition of the eccentricities and the inclinations depends in each case on the resolution of an algebraic equation of a degree equal to the number of planets under consideration, and involving their masses in indeterminate forms. Lagrange showed that all the roots of the equation must be real and unequal, in order that the corresponding eccentricity or inclination should oscillate between fixed limits. By giving the masses particular values (such as were then known), Lagrange showed that the above condition was fulfilled. Laplace demonstrated that the roots are real and unequal, without employing particular values for the masses. Lagrange was the first to deduce the numerical values of the limits of the eccentricities and inclinations. To accomplish this it is necessary to resolve an algebraic equation of a degree equal to the number of planets considered. To simplify this work he divided the planets into two groups, Saturn and Jupiter (to which he subsequently added Uranus), constituting one, and Mercury, Venus, the Earth, and Mars, constituting the other. After more accurate values of the masses of the planets and the elements of their orbits became known, a desire was felt for a new determination of the inferior and superior limits of the eccentricities and inclinations. Pontécoulant undertook to supply the necessary calculation, and he published his researches in his Système du Monde ; t but the numerical value of the constants which he obtained are erroneous, owing to his employing a sufficient number of decimals in his coinputations. This labor did not, therefore, extend our knowledge of this important subject. In 1839 M. Leverrier calculated anew the secular variations of thc planetary elements, and his results have since been used for correcting most of the planetary * Mém. Acad. des Sciences, 1784. + Tome iii., livre vi., chap. xi. 1834. elements and masses. Leverrier employed in his investigations the whole seven principal planets then known, , A very important contribution to this department of physical astronomy is that recently published by Mr. John N, Stockwell, in the " Smithsonian Contributions to Knowledge." In this memoir Mr. Stockwell has considered the simultaneous action of the eight principal planets, Mercury, Venus, the Earth, Mars, Jupiter, Saturn, Uranus, and Neptune, which necessitated the reduction of an equation of the 8th degree, “The equation of the eighth degree, when completely developed, is composed of eighty thousand six hundred and forty distinct monomial terms, each of which contains eight factors. The actual formation of this equation (by elimination) could therefore with difficulty be brought within the compass of an ordinary lifetime; and we must, therefore, seek for a shorter and more expeditious method of attaining results, which seem to necessarily involve such an immense expenditure of labor."* Mr. Stockwell has, accordingly, devised a most elegant method of approximation by means of which he was enabled to obtain accurate results. u sports .., , is fiiii ii We shall now briefly state the important conclusions at which he has arrived. For Mercury's orbit, the limits of the eccentricity are 0.1214943 and 0.2317185. The perihelion completes a revolution in the heavens in 237,197 years. ,, The limits of the inclination of its orbit to the fixed ecliptic of 1950, are 10° 36' 20" and 3° 47' 8". For Venus the limits of the eccentricity are 0 and 0.0706329. Since the theoretical eccentricity of the orbit of Venus is a vanishing element, the perihelion of her orbit can have no mean motion." The inclination of her orbit to the ecliptic of 1850 varies between the limits 0 0' 0" and 4° 51'0".'.' The elements pertaining to the motions of the earth being of greater interest to us than those pertaining to any of the planets, he has given us more details for this case. The eccentricity varies between the limits 0 and 0.0693888. The mean motion of the perihelion is indeterminate. The inclination of the apparent ecliptic to the fixed ecliptic of 1850 is always less than 4° 41. The mean value of the precession of the equinoxes in a Julian year (365.25 days) is equal to 50".438239, where the equinoxes' will perform a complete revolution in 25,694 years.' Owing to the secular inequalities of their motion, this time may vary from the actual time by 281 years. The tropical year, owing to the variation of the rate of precession, máy be larger than at present by 49.27 seconds; and it may be shorter by 59.13 seconds. The tropical year is now shorter than in the time of Hipparchus by 11.3 seconds. * Memoir, p. 12. The mean value of the obliquity of both. the apparent and the fixed ecliptic to the eqnator is 23° 17" 17". The limits of the apparent ecliptic to the equator are 24° 35' 58" and 21° 58' 36". These limits are in part dependent on the spheroidal figure of the earth. The duration of the seasons is greatly modified by the eccentricity of the earth's orbit. ' At présent the difference between summer and winter is about 7.75 days; but when the eccentricity is a maximum, or nearly that, and the transversé axis passes through the solstices, both of which conditions have been fulfilled in past ages, the summer in our hemisphere will be about 198.75 days in length, and the winter will have a length of only 166.5 days, giving a difference of 32.25 days, or more than a month. The distance of the earth from the sun will vary in the course of a year by 13,000,000 miles, or by nearly a seventh part. i. The limits of the elements of the other planets are as follows: Mars, 7 max. eccentricity, 0.1396550, min. 0.018475 0.0843289, 0.0123719 Uranus, 0.0779652, “ 0.0117576 Neptune, . " 0.0145066, 0,0055729 “ Mars, -- "max. inclination, :7° 28' 0" 6". 0° 0' 0" . Jupiter, 16', o 0 28 56 1 0 14. 23 The mean motion of the perihelia of the orbits of Jupiter and Uranus is just the same for each, 3”.716607. The motion of the nodes of Jupiter's and Saturn's orbit, on the invariable plane of the solar system, is the same, being 25".934567, in a retrograde direction. These relations Mr. Stockwell embodies in two general propositions, as follows : “I. The mean motion of Jupiter's perihelion is exactly equal to the mean motion of the perihelion of Uranus, and the mean longitudes of these perihelia differ by, exactly 180°. II. The mean motion of Jupiter's node on the invariable plane is exactly equal to that of Saturn, and the mean longitude of these nodes differ by exactly 180°,"* His investigations bring to light a curious relation between the mass of Venus and the maximum value of her eccentricity. If the mass be gradually increased, the maximum value of the eccentricity would be increased, until the mass reaches a certain limit, when the maximum value of the eccentricity will diminish with a further increase of the mass. A similar relation seems to hold in the case of the earth. Whether this be an accidental relation or not, is uncertain. It has the appearance of a real physical significance. , ! . . . . Art, V.-1. The University of Pennsylvania, Public Inau guration of the New Building erected for the Department of Art and the Department of Science. Newspaper Re port. Philadelphia, 1872. .. . 2. University of Pennsylvania. Special Announcement of the Organization and Courses of Study of the New Department of Science. To be opened September, 1872. Pamphlet. Philadelphia, 1872. THERE are a great many well-meaning people-probably the majority —who think that critics are in their glory only when passing censure. Were this true, it would follow that * Memoir, p. xiv.. |