FIG. 2. As to accuracy, I find that, even with this experimental instrument, in many respects provisional in its construction, with an object glass of very poor definition (of 12 inches aperture and 25 inches focus), employed in the open air with no protection whatever from the wind, and placed merely upon an ordinary table, or box of sand, I obtain results of at least equal precision with those given by transit instruments of about the same size, under the best circumstances. The probable error of the clock correction found by this instrument, from a set of stars such as would ordinarily be taken with a transit, for clock and instrumental corrections, is not over 10.05 sec. From my practical experience with transits in actual work, I do not hesitate to assert the belief that for instruments of moderate size, this new construction is more serviceable than the meridian transit. The second instrument to be described is also an equal altitude instrument, but is constructed on a totally distinct principle, and is of a very different order of accuracy. It is intended to supply the place of a sextant for land use, and of small, portable transits in the determination of time by surveyors, exploring parties, watchmakers, and all who desire a simple means of finding the time within a second. It consists of a swinging bar, suspended at the upper end on a pivot, in such a way as to permit the bar to assume freely a vertical position, without any torsional revolution. To the bar is affixed a small telescope, the object glass near the bottom, the eye-lens at the top of the bar, with a horizontal wire in the common focus. Below the object glass a frame is fixed to the bar, carrying a plane mirror swinging on a horizontal axis, and provided with a clamp for fixing the mirror at any desired inclination. Below this is a metal bob. The whole construction thus forms a pendulum, which is suspended inside a large tube, at the base of which are approximate levelling screws. (FIG. 2.) It is evident that if the instrument be turned towards the sun, and the mirror revolved until rays pass after reflection directly up the telescope tube, an eye looking into the telescope will see an image of the sun in the field which, as the sun rises or falls in altitude, will appear to cross the horizontal wire, which, in fact, represents a section of a small circle in the heavens, parallel to the horizon. This brief explanation will indicate clearly the use of this instrument in getting the time by equal altitudes of the sun. For this purpose it possesses the most surprising facility and accuracy. The instrument can be adjusted in a few seconds so as to bring the sun to the proper place in the field. The mirror being clamped, the transit of both limbs of the sun over the horizontal wire may be observed to within a half a second by an ordinary watch. The instrument is then set aside until a corresponding time, afternoon, or next morning, when the transit is again observed. The watch error from noon, or midnight, of course follows by applying to the mean of the observed times, the "equation of equal altitudes, and the "equation of time," as explained in the text book. For cheapness and simplicity, this instrument has, of course, very greatly the advantage over the sextant with the artificial horizon, and over the small portable transits used for this purpose by watchmakers and others; while in point of accuracy it is equal or superior to them. By an extended series of tests made on twelve instruments of this construction, I find the probable error of a single determination of time to be 0.8 sec., which could be somewhat reduced by employing three wires instead of one. The degree of accuracy already attained is, however, sufficient for the purposes of those for whose use the instrument is intended. TIDAL THEORY OF THE FORMS OF COMETS. By GEO. W. COAKLEY,1 of New York, N. Y. 1. It 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, 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 CBp, and let the distance of a particle of the comet at B from the sun be BS71; 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 (1-2 cos 0+). Letß, then r1r (1-2 ẞ cos 0+2). = Also r, sin y=p sin 0, .. sin y sin 0 (1-2 B cos 0+32)-1, (1—2 r (1-2 8 cos 0+82)=(1- cos 0) (1-28 cos 0+2)-4. 5. Let the unit of mass be that of the sun, and let the comet's mass be denoted by μ. Then the accelerative attraction of the comet on one of its particles B is fi= The accelerative attraction of the sun on every particle of the comet, considered as μ |