confining ourselves to those which have appeared in the course of 2,000 years, a minute in the probable duration of the solar system! But our calculations have reference only to those which approach the earth near enough to become visible. It remains to estimate the probable number which traverse our system at all possible distances, when we shall arrive at numbers so great that they will justify the expression used by Kepler. We shall follow as our authorities Lambert and Arago, modifying according to the state of the facts in the present day the figures employed by them in their estimation of the number of comets within our system. Lambert relies for his values upon the elements of the twenty-four comets of Halley's table. In the first place, he reduces their number to twenty-one, on account of the two re-appearances amongst them; in the next, he considers the position of the perihelia, two of which exceed the orbit of the earth; two are situated between the earth and Venus, twelve between Venus and Mercury, and, lastly, six between Mercury and the sun. These numbers, according to him, are in accord with the hypothesis that comets are uniformly distributed throughout the interplanetary spaces. But, we may ask, according to what law is the number of known comets found to increase? At first sight it would seem that it should increase in the ratio of the spaces included within the spheres of the different planets; that is to say, proportionally to the cube of the distance. But Lambert assumes that comets are disposed in such a manner that they never meet or disturb each other in their movements. To effect this their orbits must not intersect each other anywhere; further, these orbits are not to be regarded as geometric lines, but are to include as much of the sphere of activity of each comet as will prevent incursions into the spheres of others, and avoid the disorders which would result from these incursions.' From this restriction-deduced from

the principle of final causes, and which at the present time seems to us quite unjustifiable, as, observations having proved the intersection of the orbit, the resulting perturbations are perfectly possible-the celebrated mathematician reduces the increase of the number of coinets to the ratio of the square of the distance. The numbers six and seventeen, which in Halley's table give the comets comprised within the spheres of Mercury and Venus respectively-numbers which are nearly in the relation of one to three-in his opinion justified this hypothesis. Taking, then, as our basis the comet of 1680, whose perihelion was more than sixty times nearer the sun than Mercury, Lambert came to the conclusion that the sphere of the orbit of this planet may contain sixty times sixty, or 3,600 comets. Considering, then, the orbit of Saturn, whose radius is equal to twenty-four times the radius of the orbit of Mercury, he multiplies by 600 the preceding number, and finds there would be more than two millions of comets moving within this sphere.

If at the present time we were to make a similar calculation, it would be, in the first place, necessary to double our fundamental number, since in addition to the comet of 1680 we have the comet of 1843, whose perihelion distance is equally small, and in the next place, we should have to extend the limits so as to include the planet Neptune. Under these conditions we should find not less than 45,500,000 comets!

Arago, after discussing the elements of the comets contained in the catalogues at the time when he wrote his Astronomie Populaire, adopted the same fundamental principle as Lambert; that is to say, he assumed the uniform distribution of comets within the space included by the solar system. At all events, he says, 'no physical reason can be advanced for assuming the contrary.' But, with reason, he rejects Lambert's second principle, which restricts the increase of comets

to the ratio of the squares of the distances; he adopts the hypothesis that the increase is as the cube. Now, in the catalogue of 1853 there are thirty-seven perihelia whose distances from the sun are less than the radius of the orbit of Mercury. It will be necessary, therefore, he says, to make this proportion:

13 is to 783 as 37 is to the number required;

or, performing the operations indicated,

1 is to 474,552 as 37 is to 17,558,424.

Thus, within the orbit of Neptune the solar system would be traversed by seventeen and a half millions of comets. A similar calculation, now that the number of comets whose perihelion distances are inferior to the distance of Mercury amounts to forty-three, would give more than twenty millions

of comets.

We are unwilling to leave the subject without appending to the values we have just recorded a few reflections which may enable the reader to better appreciate their import.

Everyone will readily comprehend that the question is indeterminate, and that its approximate solution can never do more than assign an inferior limit of the number required. Admitting the uniform distribution of comets in space as a probable fact, it will be seen at once that the result of the calculation will depend solely upon our fundamental number, as, for example, on the number of comets that pass between the sun and Mercury. Now, the number we have taken is evidently much inferior to the number of comets which in reality have penetrated this region of space in historic times. If for 2,000 years observation and research had been carried on as during the last two centuries, what numbers of distinct comets would not our catalogues contain! More than that, in each century the number, not counting re-apparitions, would

continue to increase, and the preceding values would rise in a similar proportion.

Besides, why limit the space by the orbit of Neptune? Is it not evident that the sphere of the comets which have gravitated at least once round the sun must extend to all those regions of the heavens where the attraction of his mass preponderates? Let us suppose that stars of the first magnitude have masses nearly equal, on the average, to that of the sun, and that they are nearly equally distributed over the sphere whose radius is equal to the mean distance of Alpha Centauri; the action of the sun would extend to the half of this distance; that is, to about 100,000 times the radius of the earth's orbit. Every comet penetrating within this distance would fall under the dominion of our system and gravitate around its central luminary in an orbit whose elements would depend upon its initial velocity.

If we extend to a sphere of these dimensions the calculation we have made for a sphere extending to Neptune, does the reader foresee in what enormous proportion the results of our previous calculation will be multiplied? It would be in the proportion of the cube of 30 to the cube of 100,000; that is to say, it would be multiplied by thirty-seven thousand millions. So that instead of obtaining the already great number of twenty millions of comets we should arrive at the stupendous number of 74,000,000,000,000,000, or seventy-four thousand billions of comets, as the minimum number of those which have each been submitted for one at least of their periods to the empire of the sun!

In presence of such considerations the comparison of Kepler is no longer a metaphor, and we are permitted to say literally with the great astronomer of the sixteenth century: 'Comets are as numerous in the heavens as fishes in the ocean.'

SECTION II.

COMETS WITH HYPERBOLIC ORBITS.

Do all comets belong to the solar system?-Orbits which are clearly hyperbolicOpinion of Laplace with regard to the rarity of hyperbolic comets-Are there any comets which really describe parabolas ?-First glance at the origin of comets. Do all the comets which have been observed up to the present time belong to the solar system? Or, as we have already suggested, are there comets which visit the sun but once, and which before penetrating to the sphere of his activity and submitting to the influence of his attraction were altogether strangers to our system?

Theoretically speaking the reply is not doubtful. A celestial body, describing under the influence of gravitation an orbit of which the sun is the focus, may move in a parabola, an ellipse, or an hyperbola. All depends upon its velocity at any one given point of its course, that is, upon the relation existing between the velocity and the intensity of gravitation at that point. The better to explain this let us take a point whose distance from the sun is equal to the mean distance of the earth, and let us suppose the body to have arrived at this point. For certain velocities, which we may call elliptic or planetary velocities, the orbit described will be either a circle or an ellipse; for a greater velocity (equal to the mean velocity of the earth multiplied by the number 1.414, that is by the square root of 2), the curve will be a parabola, with endless branches; for a