Observations were also made on the Tummel at Ballinluig, on the Dochart and other feeding streams of Loch Tay at Killin, on the Forss in the north-west of Caithness, in the sea at Scrabster and at Wick in Caithness, on the Wick river, on the Glass in Strathglass, the Eden near Cupar Fife, on the Nith estuary and on Lochrutton, Kirkcudbrightshire ; but from irregular hours of observation, short period of observing, or other causes, they have not been reduced to the form of curves. A number of observers in various parts of Ireland were supplied with instruments, but only two sent in reports of work done. SEA AT MOVILLE. One of the records was an admirable series of observations in the sea at Moville, on Lough Foyle, by Mr. J. Lowry, from January 1889 to May 1890. 1889 CURVE XXVII.-Sea at Moville, Ireland. 9 A.M. Air Water 1390 Jan Feb Mar Apr May June July Aug. Sep Oct Nov Dec Jan Feb Mar Apr. M 601 50 40 Monthly Means of Temperature Observations on Lough Foyle at Moville. Curve XXVII. gives the weekly means and affords an interesting comparison with the Bristol Channel observations. The water was colder than the air while the temperature was rising and at the seasonal maximum, but warmer than the air when the temperature was falling and at the seasonal minimum. The range of sea-temperature was decidedly less than that of air-temperature, and the curve for the water is more uniform than that for the air, showing little tendency to follow sudden and temporary changes. BELVEDERE LAKE. Mr. J. Bayliss made observations on the east side of Belvedere Lake from January 1889 to January 1890. Belvedere Lake is a small sheet of water three miles south-west of Mullingar, in West Meath. The temperature of the water (Curve XXVIII.) was almost always higher than that of the air, and had a nearly equal range but was less subject to small irregularities. Monthly Means of Temperature Observations on Belvedere Lake, CURVE XXVIII.-Belvedere Lake, West Meath, Ireland. 10 Jan Feb Mar Apl May June July Aug Sep Oct Nov Dec J 60 501 40 CURVES. Weekly means of water- and air-temperature at twentyeight stations. The curves are drawn on the same scale, the entries corresponding to the average temperature (usually at 9 A.M.) for the week ending each Saturday. The temperature readings are corrected for instrumental errors, but in several instances the readings are subject to uncertainty sometimes on account of the observer only reading to whole degrees or to half or quarter degrees. On the Capture of Comets by Planets, especially their Capture by Jupiter. By Professor H. A. NEWTON. [A communication ordered by the General Committee to be printed in extenso 2 among the Reports.] 1. SOME years ago I obtained and published a formula expressing in simple terms the total result of the action of a planet in increasing or diminishing the velocity of a comet or small body that passes near the planet. This formula is practically a modification of the integral of energy, the smaller terms in the perturbing function being omitted. A very brief and partial treatment of it was presented to this Association in 1879 at its Sheffield meeting. Within the last two or three years several astronomers have made special study of the manner of Jupiter's action in changing the orbits of comets that pass very near him. M. Tisserand has given us an expression connecting the major axis, inclination, and parameter of the orbit described before coming near to Jupiter with the corresponding elements of the orbit after leaving the neighbourhood of the planet.3 M. Schulhof has applied the formula of M. Tisserand as a criterion for determining the possible identity of various comets whose orbits pass near to Jupiter's orbit. Messrs. Seeliger, Callandreau, and others have continued these investigations. The interest thus shown in the problem has led me to resume the study of the subject and to work out the results of the formula obtained by me in 1878 more fully than they have been hitherto developed. 2. One of the remarkable distinctions between the comets of long (or infinite) periods and those of short periods is that the orbits of the latter have almost without exception direct motions and small inclinations to the plane of the ecliptic, while the orbits of the former have all possible inclinations between 0° and 180°. At first sight this seems to imply that the two groups of comets are radically distinct in origin or nature one from the other. The most natural line of investigation therefore is the effect of perturbations in bringing or not bringing the comets to move with the planet after the perturbation. 3. The algebraic processes by which was obtained the formula for the change of energy which a small body experiences from passing near a planet were given in the article cited, and they need not be here reproduced. The following was the resulting equation, viz. : and it was obtained from the general differential equations of motion by making assumptions not greatly differing from those used in obtaining Laplace's well-known theorem, that a sphere of suitable magnitude may be described about the planet as a centre, and that for a tolerable first approximation the comet may be regarded as moving when without American Journal of Science, III., vol. xvi., 1878, p. 175. 2 Report, 1879, p. 274. 'Sur la théorie de la capture des comètes périodiques,' Bull. Astron., tome vi., juin et juillet, 1889. ✦ 'Notes sur quelques comètes à courte période,' Astron. Nachrichten, No. 2964. this sphere in a conic section of which the sun is the focus, and as moving when within the sphere in a conic section (an hyperbola) of which the planet is the focus. In other words, only perturbations of the first order of magnitude are taken account of. A comet is treated throughout this paper as a small indivisible body whose mass may be neglected. 4. Notation. The symbols used in (1) and also other symbols which I shall have occasion to use may be thus defined : : Let C, be the orbit of the comet about the sun before the comet comes under the appreciable action of the planet; the orbit of the comet about the sun after perturbation by the planet; C the hyperbolic orbit of the comet relative to Jupiter when near the planet; the elliptic orbit of Jupiter about the sun; A the point on C, which is nearest to I; h E the point on which is nearest to C,; d the length of the straight line EA, being the perpendicular distance between the orbits at their nearest approach; w the angle between the tangent of C, at A and the tangent to I at E; m a f the distance which the planet has yet to pass over to reach E when the comet is at A (h may be negative) ; the mass of the planet; sun's mass= unity; the unit of distance, in general the mean distance of the earth from the sun; the sun's attractive force at the unit of distance; v, the planet's velocity in its orbit at E; to the comet's velocity in its orbit C when the comet enters the sphere of Jupiter's perceptible influence; 8 the comet's velocity at A relative to the sun; @, the semi-axis major of C, (negative if C, is an hyperbola); the angle between the tangent to at O (drawn in the direction of the planet's motion) and the line from the planet to the vertices and centre of C; A the semi-transverse axis of C; B the semi-conjugate axis of C (hence equal to p); the distance of the planet from the sun; Τι the distance of the comet from the sun; O the distance of the comet from the planet; P, and p distances of the comet from the sun at selected epochs before and after perturbation; u, and u the velocities of the comet at the selected epochs ; Δ the increase to which v2_2fa2_ 2mfa2 2fa2_2mfa2 receives by the planet's Το action during the whole period in which the comet is passing near to Jupiter. 5. If we assume two epochs, one before and one after the perturbation, at which the comet is equally distant from the planet, the term 2mfa2/r. is the same at both instants, and it disappears from the value of A. Therefore But by the well-known formulas from the law of gravitation, P This equation is valid whatever be @, the major axis of the orbit C, and may be used to determine the major axis of either orbit from the elements of the other. My present purpose is, however, to study the action of Jupiter in changing orbits that are originally parabolas, and hence in general @, will be taken infinite. In that case It will be found that the second member of (2) depends on w, d, and h, and these are known quantities when the elements of C, and I are given. The use of the equation is, moreover, greatly simplified and enhanced by the fact that the plane of the planet's orbit is involved only in so far as that it must contain the tangent to I at E. 6. In the second member of (2) all the factors are positive except cos ; hence if <, @ is positive and the orbit C is an ellipse; but if >, @ is negative and is an hyperbola. This result may be thus expressed: If the comet passes in front of Jupiter the kinetic energy of the comet is diminished; if it passes behind the planet the kinetic energy of the comet is increased. The reason for this may also be given in general language. If the comet passes in front of the planet the comet's attraction increases the velocity, and hence increases the kinetic energy of the planet, and vice versa. But the total energy of the two bodies is constant, so that when that of the planet is increased that of the comet is diminished, and vice versa. 7. It is desirable now to transform the value of @ given in equation (2) so as to be able to determine the major axis of the new orbit of the comet directly from the circumstances of its initial approach to the planet before perturbation; in other words, to find @ in terms of «, d, and h. For this we must find in terms of w, d, and h values for s, p, a, and . 1891. Ꮮ Ꮮ |