After this long preamble, a very few words will suffice to shew how, without any observation, of the phases, or any derived from polarization, it is possible to ascertain that comets shine with a borrowed light.

I have, in truth, but recently established (p. 23), that a body luminous of itself ought to have, whether to the eye or in any given telescope, precisely the same brightness, whatever be the distance at which it is placed in relation to the observer. I have also proved, on the other hand, that the visibility of a body does not depend on the angle which it subtends, at least when that angle does not descend below certain limits. So much being established, it only remains for us experimentally to resolve the following questions: In what manner does a comet disappear? Is this disappearance the consequence of an excessive diminution in the apparent dimensions of the star, arising from a vast increase of its distance from the earth? May it not rather be attributed to a change in its intensity? Well, then, all astronomers will answer that this last cause of disappearing is the true one. The most part of observed comets, and that of 1680 in particular, have disappeared by a gradual enfeebling of their light. They have become, so to speak, extinguished. The evening of the day in which we cease to be able to observe them, they still subtend angles which are very considerable. But this mode of disappearance, I have proved in detail, is irreconcileable with the existence of an inherent light, and comets there. fore borrow their light from the sun.

In the various experiments which have led the way to this conclusion, we have admitted that, during its variations of distance, the luminous body under our observation did not change in its physical constitution; but, it is true that comets do not answer to this description. There is here, then, a substantial objection, and some short reflections will therefore be required.

Till lately, it was very generally believed that the nebulous matter of a comet gradually was condensed, in proportion as in its elliptical course it was removed from the sun. This con

densation could not fail to produce in the star a brightness superior to that which without this change it could have maintained.

Observation has shewn us that this star becomes more and more faint, even in the alternative of a theory, based upon the hypothesis that an unchanging constitution would indicate a constant light. The real increase of intensity which would have resulted from the supposed condensation of the nebulous matter, was calculated, therefore, to exhibit more strikingly the disagreement between the calculation and the experiment. It ought to add strength to the conclusion to which this disparagement has conducted us; and so, in our argument, we might legitimately have abstracted the alleged condensation which the nebulosity of the comet was supposed to undergo. Now-a-days, however, on the contrary, it is proved that, instead of contracting, the nebulosity dilates, as it withdraws from the sun. I would not venture now, as I did formerly in my public lectures, to conclude without more ado, that because the light of comets became gradually more faint, therefore their light is a reflected light. It is necessary, in addition, to take into account the scattering which the nebulous matter undergoes. It is necessary to demonstrate, that the real diminution of intensity which thence results is not sufficient to explain how, sooner or later, the most brilliant comets disappear. But this appears neither difficult nor complicated, as the reader shall judge.

Up to the present time, the most brilliant comets have ceased to be visible from the earth, as soon as, in their progress round the sun, they have removed themselves from this luminary to a distance equal to a radius of the orbit of Jupiter; that is to say, to five times the radius of the almost circular curve which the earth annually describes. Let us now then consider a comet which, like that of 1680, had its perihelion within the orbit of Venus. According to the calculations of M. Valz, the real diameter of its nebulosity would augment with its distances from the sun, according to the following progression:-At the distance of Venus, 10; the Earth, 29; Mars, 76; Ceres, 173; Jupiter, 278.

This progression of diameters differs but little from the following:-1; 3; 8; 17; 28.

The quantity of nebulous matter which, at the distance of Venus, occupies a spherical volume of a diameter equal to 1,

will therefore be expanded into volumes of the same form 3, 8, 17, 28 times larger at the distance of the Earth, Mars, Ceres,

and Jupiter.

These spheres, diaphanous on account of their extreme length, exhibit themselves in simple circles. It is in the apparent surface of these circles that the same quantity of nebulous particles appear successively sprinkled with greater or less uniformity. The luminous intensity of the nebulosity will evidently vary in the inverse ratio of its density,-will follow the law of the surface of circles; in other words, that of the squares of their diameters, or the squares of the numbers 1, 3, 8, 17, 28.

I have already proved, that a comet which had inherent light would not experience, at any distance in which it could be observed, any other variations of density than those of which we have already specified both the cause and the law. We have now, therefore, in addition, only experimentally to examine if these variations are sufficient to render the most brilliant comets invisible, as soon as they have reached the orbit of Jupiter. The following is the method in which the experiment should be undertaken.

A telescope should be selected which had a large opening, and a feeble magnifying power, and with which the comet could be observed during the whole duration of its appearance. So much being ready, the day, for example, in which the star shall be found distant from the sun, a space equal to a radius of the orbit of Venus, it should be examined, first, as a point of comparison, with the most feeble magnifier, and then with magnifiers 3, 8, 17, 28 times stronger. During these experiments, the same quantity of light, that, namely, which the unvarying extent of the object-glass would admit of,—that, in short, which the circular image of the comet depicted in the first experiment, would be found successively spread out upon circles 3 times, 8, 17, 28 times, larger diameters than in the first of the observations. But is it not evident, that the diminutions of intensity which these artificial dilatations will induce upon the luminous matter of the comet, will be respectively equal to those which result from the corresponding natural dilatations, which the star undergoes in removing from the sun? or, in other terms, that the simple changes in the eye-glass cause the comet to pass, so

to speak, in a few seconds, from the distance of Venus to that of the Earth, of Mars, Ceres, and Jupiter? If this, then, be true, and seeing the comet with our telescope supplied with its weakest magnifying powers when it crosses the orbit of Venus, we subsequently examine it in succession with the help of a magnifying power 3 times, 8, 17, 28 times stronger; then, if it be always seen, it ought to appear the same as when seen with the first and feeblest magnifying power, at the epochs in which its proper movement shall have transported it to distances from the sun, equal to the radii of the orbits of the Earth, of Mars, Ceres, and Jupiter; and if it be not seen, for example, when it has reached the orbit of Jupiter, it is not only because it has been subjected to the enfeebling which might result from the scattering of the matter of which it is formed;-it is because it does not act as a body which has an inherent light; it is because it borrows its brightness from the sun!

It is true that all comets are not equally eligible for these kinds of experiments. We ought to select in preference those comets without apparent nucleus, and without a tail, because they appear to be less subject than others to sudden and irregular changes of figure, and because that, during the act of the singular dilatation they undergo in removing from the sun, and of which Mr Valz has given the law, it is probable that all the parts, from the centre to the circumference, undergo analogous changes. Without this condition, the natural dilatation of the nebulosity cannot be assimilated to that which we obtain artificially in the foregoing proof by means of the eye-glass of the telescope. The importance of this remark will be perceived, when I notice that, in the comet of 1770, the nucleus and the nebulosity properly so called, were very far indeed from experiencing proportional changes

* In proof of this assertion, I subjoin the measurements which Messier has given both for the nucleus and nebulosity of the comet 1770:

On the 17th June 1770, the nucleus measured 0.22, the nebulosity 5.23

The method which I have now described so much at length, is exposed, I believe, to only one kind of objection; it may be imagined that the matter of which the comet is composed has no inherent light of its own, but that it acquires it when under the action of the solar rays.

This hypothesis, in reality, would be nothing more than the reproduction of that system which Euler has developed in his "Letters to a German Princess ;" and, according to which, the light which enables us to perceive certain bodies, such as paper, porcelain, &c. is not composed of rays which are truly reflected, but rather of a particular kind of light, which these bodies engender by being subjected to vibrations under the action of the solar rays. This, it will be perceived, is a difficulty of pure theory, and which might be as much applied to the light of the moon, of the planets and their satellites, as to that of comets. But the sole object which I proposed to myself in this paper, was to discover a method which would enable us to decide whether these bodies were to be ranged, as it regarded their luminous properties, in the same category as our own globe, or as Mars and Jupiter, &c. That other question, whether the light which enables us to perceive coloured bodies is reflected, as Newton supposed, at the surface of very fine material plates, or proceeds from vibrations communicated to the air by the constituent particles of bodies; this question, I say, is of a very different kind and extent, and this is not now the place to discuss it.

3. On the Comet which will pass its Perihelion in November 1835.

In the Annuaire of 1832 I published a long notice, in which there is an elementary explanation of all that astronomy at the present day possesses of precise and mathematical information on the motion and nature of comets. To that treatise I beg to refer the reader. In the present note I shall content myself with mentioning some modifications made on the former results, pointing out the principal circumstances which may render this sixth appearance of the same comet useful to science, and indicating the constellations in which we are to seek it.

After having determined, by the assistance of very laborious