TIDAL THEORY OF THE FORMS OF COMETS. BY GEO. W. COAKLEY,' of New York, N. Y. 1. Ir is proposed in this paper to inquire, whether the forms and transformations of comets may not be explained by the mutual gravitation of their parts, together with a tidal disturbing force, due to the difference of the sun's attraction on the several parts of a comet. For this purpose a comet will be taken to be a mass of gaseous matter, large enough to give it a certain controlling power over its figure of equilibrium, especially when far away from the sun. But, on the other hand, the mass will be taken to be so small, that the comet's figure of equilibrium is readily and largely disturbed, as it approaches or recedes from the sun. 2. The success of La Place's Dynamical Theory of the Tides produced in the waters of the Earth, by the disturbing forces of the sun and moon, depends mainly upon the smallness of the ocean depths. Hence it will be found not very suitable to the discussion of those great changes of figure produced throughout the whole dimensions of a comet, which a similar disturbing force of the sun may produce. On this account I have preferred to apply to the case of the comets what has been called the Equilibrium Theory of the Tides. 3. There exist three model treatises on the equilibrium theory of the tides, all included in the Glasgow edition of Newton's Principia, published in 1833. They are by Daniel Bernoulli, MacLaurin, and Euler; the first in French, the other two in Latin. Bernoulli's treatise is too closely restricted to the special case of the Earth's tides by the condition that the changes of figure, produced by the disturbing forces, shall always be small. This condition, with more or less stringency, is employed in all these treatises. But the mode of treating the subject, and of dealing with the disturbing forces, adopted by Euler, seems less restricted to this condition; and therefore more suitable to the case of the comets, than the methods employed in the other two treatises. I have not, therefore, hesitated to make Euler's plan, in a measure at least, my model in treating the disturbing force of the sun, or 1 Professor of Mathematics and Astronomy, University of the City of New York. its tidal action in producing and changing a comet's figure of equilibrium. I have, however, varied somewhat the mode of resolving the sun's disturbing force, and perhaps the whole subject will be found to be treated in a more strictly analytical manner than that adopted by Euler. I have not, however, been able to carry the analysis to its last degree of completeness, partly on account of the law of a comet's density being unknown; and also from the fact that the equations of equilibrium depend upon the comet's form, while the varying form depends again upon the forces expressed by these equations. La Place notices this difficulty in the problem of the Earth's figure of equilibrium, and is only able to deal with it successfully by the condition, that the Earth's figure never departs widely from that of a sphere. But in the case of the comets, there is generally a wide departure of their figure of equilibrium from that of a sphere, or from any given figure at a given time, in consequence of their great change of distance from the sun, and hence of a great change of his disturbing force. This fact, of the large and ever-varying tidal disturbing force of the sun on a comet's figure of equilibrium, makes the complete analysis perhaps the most difficult problem of celestial mechanics. I have, therefore, only attempted the solution so far as to give what might be called qualitative, in contradistinction to quantitative results, except in a few special cases where the formulæ obtained seemed applicable to at least an approximate determination of the limits of the masses of certain comets, for which suitable data could be obtained from observation. 4. In the accompanying figure on page 161, let SK represent the transverse axis of a comet's orbit, S the position of the sun's centre; let C be the comet's centre of gravity, at such a distance, r=SC, from the sun, in the first instance, that the comet has nearly or quite a spherical figure, the section of which by the plane of the orbit is the circle BBB. Let the radius of this sphere be CB=p, and let the distance of a particle of the comet at B from the sun be BS=1; and let the radius CB make the angle BCS=0 with the straight line joining the centres of the sun and comet, while BS makes the angle BSC with the same line. = If C be taken as the origin of rectangular co-ordinates in the plane of the orbit, CS the positive direction of the axis of x, and CB that of the axis of y, then CA, and AB, the co-ordinates of x=p cos 0, y=p sin 0, and we also evidently have Hence r(r−x)2 + y2 = r2+x2+ y2—2rx-r2-2rp cos 0+p2 ... r2 = r2 (1—22 cos 0+). Letẞ, then r1r (1-2 cos 0+2). Also r1 sin y=p sin 0, .. sin yẞ sin 0 (1-2 ẞ cos 0+32)—1, and 5. Let the unit of mass be that of the sun, and let the comet's mass be denoted by . Then the accelerative attraction of the comet on one of its particles B is fi The accelerative at traction of the sun on every particle of the comet, considered as = all condensed into its centre of gravity at C, or the force which produces the motion in the orbit, is f. But the sun's acceleration of the particle at B, in the direction of BS, is f fs=‚1⁄2 (1—2 ẞ cos 0 + 52)−1 =ƒ1⁄2 (1—2 ß cos 0 † 52)−1. = 1 or 6. Let us resolve the force f, which acts in the direction of BS, into two rectangular components, one of them, f, in the direction BD, parallel to CS, and the other, fs, in the direction BA, perpendicular to CS and to BD2. They are, f=f3 cos y, and ƒ;= £3 sin . 7. If now we suppose a force equal to f2, or to the force which, acting at the comet's centre of gravity, maintains its orbital motion, to be applied to every particle of the comet, in directions parallel to CS, but opposite thereto, or in directions parallel to CB1, away from the sun, then the comet may be supposed to be at rest, or to move only by virtue of its inertia. If then the component, f, in the direction of BD2, were equal to the force f applied in the direction BD,, the particle of the comet at B would be undisturbed, by this component, in its relation to the centre of gravity at C. But the force f1, at B, exceeds the force f by f-f2; this excess therefore tends to draw the particle at B away from C in the direction BD. Moreover, the component f tends to move the particle at B in the direction BA, or towards the line, CS, joining the centres of the sun and comet. 8. Let us again resolve these two disturbing forces, namely, f-f2, and f, each into two new components, in the two directions BD, or the prolongation of CB, and a perpendicular to BD, at B, on the side towards CS. The components of f-f2 are, fo=(f-f2) cos 0, in the direction of BD, and ƒ;=(ƒ‚—ƒ1⁄2) sin 0, perpendicular to BD. The components of Hence f are ff; cos (90°+0), in the direction of BD3, and ƒ,=ƒ1⁄2 sin (90°+0), perpendicular to BD. f=(fa cos -2) cos 0=f, cos 0 cosy (1-2 3 cos 0+5)-1-f2 cos 0 fs-f2 sin 0 sin y (1-23 cos 0 +5o)−1 ff sin 0 cosy (1-2 3 cos 0+)-f2 sin 0 faf sin cos 0 (1-2 ẞ cos 0+,52)1. 9. If therefore F, is the whole disturbing force in the direction BD, and F, the whole disturbing force at B perpendicular to BD, we have F1=f2 cos (0+y) (1—2 ♬ cos 0+32)1 —ƒ1⁄2 cos 0 } But we readily find cos (0+) = (cos 0-3) (1—2,3 cos 0+5)-4. Hence F1 = f (cos 0-3) (1-2 3 cos 0+52)—f, cos 0 or, replacing f by its value,, we have F1 = {(cos 0 — §) (1 — 2 ẞ cos 0 + 52)—1 — cos (? 1 1 2 } F. = sin 0 {(1 — 2,3 cos 0 + 52)— — 1} The particle at B is therefore urged towards the comet's centre of gravity at C by the force fi-Fi or 1 -F; and it is also urged tangentially towards CS by the force F. to 10. If ◊ be made to take all values from 0° to 360°, or from 0° 180°, and the plane of the circle at the same time be made to revolve completely around B,B1, the particle at B may have any position whatever on the surface of the comet's spherical figure. Moreover, if p be made, to vary from 0 to CB, the particle at B may be situated anywhere within the comet's figure. 11. The acceleration towards the comet's centre is μ f1-F1 = {(cos 0—,3) (1—2 ß cos 0 + 2)−1 — cos 0 1 2 P Let F' equal the value of this acceleration when applied to the particle of the comet at Bo, nearest the sun, which I shall call the comet's front; and let F," be its value at B1, the part most remote from the sun, which I shall call the rear of the comet. Then, since cos = 1 in the first case, while cos 0-1 in the second case, we have |