which, by the way, Bode's name has been improperly given, would make the distance of Neptune beyond the orbit of Uranus nineteen times the distance of the Earth from the Sun, while it is in fact less than eleven times this distance beyond it, so that the fallacy of this formula must now be so evident as to require no demonstration.

Discordances such as those which exist in the application of this law to the planetary system, would afford sufficient reason for rejecting the analogy of Kirkwood; but, with even these discordances, the fact, that a single formula would approximately represent the truth to so great an extent, would justify us in bestowing much time upon its consideration.

[Dr. Gould then gave a brief sketch of the points of connection between the nebular hypothesis and the new analogy,- showing how the one would lead to the other.]

It will be remarked that in the phrase "sphere of attraction," the word sphere is not used in its geometrical sense. Nor is a planet necessarily in the centre of its sphere of attraction, for upon the one side, as is the case with the earth, may be a planet comparatively near, and upon the other a smaller planet at a greater distance; whence it is evident that the extent of the sphere of attraction will be much less upon the former side than upon the latter.

In Mr. Walker's theory he assumes x, in the equation

to be a constant and equal to 1.365,256374, the term k

(*) 2

being nearly 2. In the following formulas I shall denote the quantities which refer to the Earth by a single accent, those referring to Mars by two, to Jupiter by four, and to Saturn by five, reserving three accents for the hypothetical planet between Mars and Jupiter. I make use of Mr. Walker's formula for the sphere of attraction as follows:-D being the diameter of the sphere, a being the mean distance of the planet, and m being its mass.

D"=√m" (√m" — Jm'

√m" + √m".

In these formulas you will perceive that every thing is known, except the mass and distance of the new planet; or the old planet, if you please. The only assumption is the truth of the nebular hypoth

esis. If we knew the values of D and D", the spheres of attraction of Jupiter and Mars, we should have two equations and but two unknown quantities, a'"' and m'"', and should thus have this planet restored by the nebular hypothesis alone.

I have assumed in the computation for the value of k, not 2, the constant, which Mr. Walker supposes it to be, not only from his calculations, but from a priori reasoning, but the mean of the values obtained from each planet, using the masses as given in the books, though affected of course, with some inaccuracy.

Now recurring to the above phenomena, let us take, for the sake of convenience,

Here all the quantities on one side are known, and we use the mass of the new planet thus obtained for the solution of the problem. This being substituted in either of the previous equations, will give us the mean distance of the planet.

Now the only question is as to the value to be adopted for Mr. Walker's constant, which it seems to me should be deduced from observation only.

In computing the value of k, I obtain for

is the quantity assumed for k, in obtaining the results of which I shall speak; these three planets being the only ones to which the formula

can be applied. It will be seen that the values obtained are most confirmatory of Mr. Walker's results. Unless we suppose the nebular matter to have been equally distributed through the solar system, we could not expect to find k absolutely constant, even if it were an approximation to the number 2.

Prof. WALKER here remarked that the constant used by Dr. Gould answered better than the constant 2.

Dr. GOULD. We do not know the extent of Mercury's influence inside its orbit, and hence cannot know the diameter of its sphere of attraction. Nor can we apply the formula to Mars or to Jupiter, for we do not know what planet may have been between them. We cannot apply it to Uranus, for we do not know its period of rotation. There remain but the three planets mentioned above.

Calculating, from the equations thus developed, the mass and distance which would belong to a planet between Mars and Jupiter, and thence, by Kirkwood's analogy, the corresponding time of rotation, we do not find it so great, that, by mere centrifugal force, the planet could have been exploded, and its mass scattered in the form of asteroids.

From a very rough computation of the place and size of the hypothetical planet, I obtain a mean distance 3.12,—and a mass 501,500. This mass is very much smaller than the mass of our earth, and would agree with the supposition of a small planet, smaller even than Venus, but would be at least equal in size to twelve or fifteen Asteroids.

This gives rise to a great many speculations, most interesting and important in their bearing upon the theory of the universe. I wish to dwell upon the fact that we neither know accurately the period of rotation, or the mass of most of the planets. The only element which is really well known, is the distance of the primary planets from the Then there is the difficulty to which I also alluded, in ascertaining the magnitude of spheres of attraction, that we cannot assume the nebulous matter to be equally dense; so that it cannot be demanded that the analogy should be very accurately expressed by any given data.

sun.

It is now extremely important that observations should be made upon the periods of the rotation of several planetary bodies, and is much to be desired, as bearing upon this problem, that those who occupy themselves with what may be called the natural history of astronomy should determine the times of rotation anew, and thus

enable us to decide upon the truth of a law, the discovery of which may be important in the history of Astronomy.

Prof. Walker made a remark on Saturday, with reference to the position to which Mr. Kirkwood will be entitled, should his theory be found true. The Section seemed surprised at this remark. I do not wish to express myself strongly, but certainly when we look back upon the labors of Kepler, who strove so many years with results so unpromising, until he discovered the laws which underlie the whole fabric of our solar system, and then turn to Mr. Kirkwood, a teacher in the interior of Pennsylvania - who without the sympathies of kindred minds, or the use of any library of magnitude - without calling even upon the aid of strict mathematical analysis - has fixed his attention upon this one problem, and investigated it in all its bearings, until after ten years of patient thought and labor, he has arrived at such a result as this we cannot but be struck with the similarity of the two cases; nor can we consider it as very derogatory to the former to speak hereafter of Kepler and Kirkwood together as the discoverers of great planetary harmonies.

Prof. WALKER followed with some remarks upon the same subject. He gave the formulas of the nebular theory upon the supposition that the substance is homogeneous, &c., and of rotation, as connected with Kirkwood's analogy. From these calculations, he derived results relative to the mode of formation of the earth from the ring, the form in which it must have existed according to that theory.

ON GEOMETRICAL INTERPRETATION OF ANALYTICAL NOTATION. BY J. PATTerson.

THIS long communication was of such nature that no abstract could be made.

ON A CURIOUS PHENOMENON RELATING TO VISION.
BY PROF. Lovering.

PLATEAU, WHEATSTONE, and FARADAY have made ingenious researches on the appearances exhibited by wheels in rotation. The optical deceptions which they describe all turn upon that organic peculiarity of the eye called the persistency of impressions on the retina. The phenomenon to which this paper relates also originates in an organic law of vision, but one wholly distinct from that just mentioned.

The phenomenon is as follows. When the plane of rotation of an ordinary windmill nearly coincides with the axis of vision of an eye which is looking towards it, we have observed that the mill will appear suddenly to shift to the other side of the axis of vision, and turn in the opposite direction. In a few minutes it will resume its first position, and then change again; and so on repeatedly as long as we continue to look steadily at it. The change in the direction of rotation is always associated with the displacement of the plane of rotation. The observer may soon satisfy himself, from the direction of the wind and otherwise, that both these changes are apparent and not real. Of course, therefore, the wheel is not seen to move from one side of the axis of vision to the other; it does not move at all. The change originates in the eye and in the judgment which the mind passes on the facts of sensation, and is not subject to the law of continuity.

Let W M represent the intersection of the place of rotation of the windmill with the horizon; let E be the place of the eye, and E W the direction of the extremity of the left horizontal arm of the mill, and E M the direction of the end of the right horizontal arm. What we wish to explain is, why the line W M, which is really in the position assigned to it in figure 1, appears sometimes to occupy the position which we have given it in figure 2.

We start with what we think must be admitted to be a general fundamental law of vision, namely, that the eye is so organized that it is able directly to see the direction of a point, but not its distance in a straight line from itself. The eye can see directly the distance of two bodies from each other, at least that resolved portion of it which is perpendicular to the mean axis of vision, because this portion subtends an angle which represents the difference between two directions. The measurement of such a distance is embraced in the provision for seeing direction. All those methods used in Geodesy and Astronomy to obtain that element of the distance which is parallel to the axis of