scope of Herschel, do not contribute more than 12 per cent. to the brightness of the Milky Way.

2nd. If the effect produced by the stars visible to the naked eye be abstracted, the illumination of the ground of the heavens in the middle of the Milky Way will be expressed by 0.92007, the total illumination being represented by unity.

3rd. The total illumination of the heavens, in the direction of the poles of the Galactic circle, is only ths of what it is in the middle of the Milky Way.

4th. When the combined lustre of stars of distinct classes is considered, the ratio of the illumination in the Milky Way to that in the direction of its poles, gradually increases as the stars are more remote.

5th. If we exclude from consideration the stars visible to the naked eye, the brightness of the heavens in the direction of the poles of the Galactic circle, is only one-sixth of the brightness of the Milky Way.

It must be admitted that the hypothesis of the extinction of light, from which M. Struve has deduced the foregoing interesting conclusions, cannot be considered as a principle of physics, the reality of which has been established beyond all doubt upon the solid basis of observation. It has been mentioned, that his reason for adopting this principle was founded on the circumstance of the space-penetrating power of the telescope having been discovered by him to be in reality much less than what theory indicated it to be. The question may be considered under the following aspect:-In the Milky Way the average number of stars visible in the field of view of the 20-feet reflecting telescope of Herschel, was found by M. Struve to be 122, and the distance of the smallest of such stars, as deducible from his researches on the law of the density of the stars, was determined by him to be 25.672 units. Hence, if the density of the stars in the direction of the plane of the Milky Way was uniform, as M. Struve in his researches assumed it to be, the number of stars visible in the telescope at the distance of 74.83 units, ought to be 3021. Now, the space-penetrating power of the telescope, as computed upon the ordinary principles, was found by M. Struve to be represented by 74.83 units. The conclusion would, therefore, appear to be unavoidable; either that the density of the stars diminishes towards the limits of the Milky Way, or that the light of the stars is gradually extinguished in its passage through the celestial regions to the earth. With respect to the question of a diminution of density, it is to be remarked that, since the inconsistency between theory and observation, above referred to, is presumed to manifest itself in every part of the contour of the Milky Way, it cannot be considered as available in explaining that inconsistency, unless the sun be supposed to be situate near the centre of the great sidereal disk, constituting the Milky Way. M. Struve, however, contends that we have no knowledge of the limits of the Milky Way-that to us it is absolutely unfathomable, and consequently that there does not exist any probability of the sun being situate near the centre of the Galactic circle. Upon these grounds, therefore, he maintains that the explanation of the inconsistency, by a diminution in the density of the sidereal stratum, in the direction of its principal plane, is inadmissible, in ipso limine.

The assertion of M. Struve with respect to the absolute unfathomability of the Milky Way, appears to be founded on certain observations cited by Sir William Herschel in the course of his sidereal researches ; but, with all due deference to the authority of the illustrious astronomer of

Pulkowa, we are inclined to believe that he has not apprehended the real drift of the language used by Herschel on those occasions to which he refers. It appears to us evident, from the tenor of Sir William Herschel's remarks on the Milky Way, scattered through his various papers, that he considered it to be a vast sidereal system of definite dimensions, and, generally speaking, of ascertainable limits. In his important paper on the subject, inserted in the Philosophical Transactions for 1785, he has remarked, that even in those cases wherein the gauges were very high, the stars were neither so small nor so crowded, as they must have been on the supposition of a much farther continuance of them, and when certainly a milky nebulous appearance must have come on *. On a subsequent occasion, indeed, he cited some observations of the Milky Way, from which it appeared that, notwithstanding the application of higher and higher degrees of optical power, there still remained traces of nebulosity in the telescope, indicating that the limits of the stratum had not been reached. It is important to remark, however, that the object he had in view in citing these observations, was not for the purpose of showing that the Milky Way was unfathomable even in those parts to which his observations referred, and to which such an expression, in a certain sense, was fairly applicable; but to demonstrate that the nebulosity in the telescope was not of an ambiguous nature-that, in fact, it was attributable to the circumstance of its consisting of stars too remote to be distinctly visible by any optical aid that was available, and not to its being in reality composed of nebulous matter†.

A more serious objection urged by M. Struve against the hypothesis of a diminution of the density of the stars in the plane of the Milky Way, is founded on the circumstance that an examination of the gauges of Herschel conducts to the same law of condensation in a direction perpendicular to that plane, with the law which he found, from the observations of Bradley and Argelander, to prevail in the immediate vicinity of the sun, as far as the stars of the eighth and ninth magnitudes. By his researches on the gauges of Herschel, he found that, at a distance from the plane of the Milky Way equal to the radius of a sphere comprehending the stars of the seventh magnitude, the density of the stars was equal to 0.41365, the mean density in the Milky Way being represented by unity. His examination of the zones of Bessel gave him 0.40525 for the density at the same distance. In like manner he found that, at a distance from the Milky Way equal to the radius of a sphere embracing the stars of the eighth magnitude, the density of the stars, as deducible from the gauges of Herschel, was represented by 0.31083; while, again, the zones of Bessel made the density at the same distance equal to 0.28410. The near agreement of the results for both classes of stars must, indeed, be regarded as very remarkable, especially when we take into consideration the very different sources from which they were in each case derived. It may be remarked, however, that the observations from which these results were derived, however trustworthy they may be in point of accuracy, can hardly be allowed to constitute a sufficiently ample basis for establishing beyond doubt so extensive a conclu

Phil. Trans., 1785, p. 247.

+ Herschel cites six of such observations in a paper inserted in the Philosophical Transactions for 1817 (pp. 325, 26, 27), and four additional observations of the same nature in a paper published in the following year (Phil. Trans., 1818, p. 463).

Etudes d'Astronomie Stellaire, p. 77.

sion as that relative to the law of the distribution of the stars. Moreover, the results of Sir John Herschel's gauges in the southern hemisphere would seem to indicate, that for the stars in the vicinity of the sun a different law of density prevails from that deduced by M. Struve *. On the other hand, the hypothesis of a gradual diminution of the density of the stars in the plane of the Milky Way, as well as in a direction perpendicular to that plane, is not only consonant to sound philosophy, but is also strongly suggested by analogy from observations of the arrangements prevailing throughout the other systems of the sidereal universe.

The other alternative suggested by M. Struve, as an explanation of the inconsistency between theory and observation, which he encountered in the course of his researches; namely, the hypothesis of a gradual extinetion of light in its passage through the celestial regions, has been objected to on very strong grounds by Sir John Herschel. If such an hypothesis were true, we might reasonably presume that, in consequence of the light being everywhere extinguished at the same distance, the Milky Way would present a uniform aspect throughout its course. As, however, observations of the actual aspect of the Milky Way do not accord with this conclusion, the hypothesis from which it is deduced is manifestly inadmissible.

It is very evident, that in the present state of sidereal astronomy the interesting question proposed by M. Struve, and discussed with so much ability by that astronomer, does not admit of a definitive solution. To attain this end it will be desirable to institute an extensive series of observations relative to the apparent distribution of the stars, both in the Milky Way and in a variety of other positions, with respect to the plane of the Galactic circle, employing at the same time telescopes of different apertures. By such means alone can we reasonably hope to arrive at reliable conclusions relative to the constitution of the great sidereal system, which presents itself to our observation under the aspect of the Milky Way, and of which the sun, as well as the greater number of the stars visible even in the most powerful telescopes, may be regarded in all probability as so many of the constituent bodies.

See the table of apparent densities for stars of different magnitudes, which Sir John Herschel has given at page 382 of his Results of Astronomical Observations at the Cape of Good Hope, &c.

APPENDIX.

I.

ILLUSTRATIONS OF PLANETARY PERTURBATION.

It is not intended here to attempt a detailed exposition of the principles of Planetary Perturbation. The object of the following remarks is mainly to elucidate the peculiar features of perturbation which characterise the mutual action of two revolving bodies, the mean motion of one of which around the central body is almost exactly double the mean motion of the other. Several examples of this mode of perturbation occur in Physical Astronomy, one of the most interesting of which relates to the mutual action of Uranus and Neptune; and it is chiefly with the view of exhibiting its influence in the theory of these planets that the series of illustrative notes, which we now submit to the attention of the reader, have been drawn up.

(1.) Let us suppose a body to receive an impulse in free space, and to be subjected at every instant to the action of a force tending to a fixed point s.

R

Let P represent the initial position of the body, and let us first suppose it to be projected in the direction PT, so that the angle SPT is obtuse. If it was not acted upon afterwards by any force, it would advance along PT with a uniform velocity depending on the intensity of the impulse; and the angle contained between the radius vector and the line PT, representing the direction of motion, would continually increase. The central force at s, however, by its incessant action, prevents the

body from moving in the same direction during any assignable interval of time, however short; so that while the body would have described the small space P Q with the velocity due to the impulse, it is in reality constrained to move in the curvilinear arc PR; and the angle contained between the direction of motion and the radius vector, instead of being equal to s QT, is now equal only to SRX. It appears, therefore, that the central force tends to make the angle contained between the direction of motion and the radius vector less than it would be, if the body had proceeded in the direction of the impulse. The same remark evidently applies to the case in which the body is impelled in the opposite direction PT, making an acute angle with the radius vector.

(2.) Since the central force acts obliquely at P, with respect to the direction of motion, it does not draw the body with its whole intensity out of the line PT. If we resolve it into two directions, one of which is parallel to the tangent PT, and the other perpendicular to that line, it will at once be seen, that only the force which acts in the latter direction is effective in pulling the body into the orbit. The intensity of this force is to that of the whole central force as the perpendicular upon the tangent to the radius vector, or as sy to s P. The remaining element of the force acts in the direction of the tangent TT, and tends to retard or accelerate the body, according as it is proceeding towards T or T', its intensity being to that of the whole force as P Y to s P.

(3.) It appears, then, that when a body is compelled to revolve in a curvilinear orbit, by the action of a force directed constantly to a fixed point, its motion is continually retarded as it recedes to a greater distance from the centre of force; while, on the other hand, when the body is approaching the centre of force, its motion is continually accelerated. It may be shewn, by a strict investigation founded on dynamical principles, that the motion of a body, when maintained under such circumstances, is so regulated, that the areas swept over by the radius vector are proportional to the times in which the corresponding curvilinear spaces are described by the body. This celebrated theorem was first discovered by Kepler, who found it to be applicable to the elliptic movements of the planets; and it was subsequently demonstrated by Newton to be true in every case of curvilinear motion depending on the action of a central force. (Principia, lib. i., prop. i.)

(4.) Although the central force at s tends to make the angle contained between the direction of motion and the radius vector less than it would have been if the body had been allowed to proceed in the direction of the impulse, it does not necessarily follow that the same angle, when considered solely with reference to the orbit in which the body revolves, will actually undergo a diminution of magnitude. Thus, although the angle SRX representing the new inclination of the path of the body to the radius vector, is less than s QT, it is not necessarily less than the angle s PT, representing the angle of inclination at P. In fact the angle contained between the line representing the direction of motion and the radius vector, or the tangential angle, as we shall hereafter for the sake of brevity denominate it, may continually increase or continually diminish, or it may even constantly retain the same magnitude; the question of its variation at any given point depending on the circumstances which determine the motion of the body at that point; namely, the relative values of the radius vector, the tangential angle, the velocity, and the force. It is manifest that if the deflection of motion at any point be exactly equal to the angular displacement of the radius vector, the tangential angle will not vary; if it be greater than the displacement of the radius vector, the tangential angle will diminish; if less, the same angle will increase.

(5.) It may be demonstrated that if the velocity of a body revolving in a curvilinear orbit under the influence of a force tending to a fixed point, be less than that of a body revolving at the same distance in a circular orbit under the influence of a force of equal intensity, directed to the centre of the circle, the tangential angle will diminish; but if the velocity be greater than that in a circle under similar circumstances, the tangential angle will increase.

(6.) If the velocity of the body at any given point be exactly equal to