Part 3 (2/2)
[9] In middle lat.i.tudes.
SECTION SECOND.
MECHANICAL ACTION OF THE MOON.
We will now proceed to give the method of determining the lat.i.tude of the axis of the vortex, at the time of its pa.s.sage over any given meridian, and at any given time. And afterwards we will give a brief abstract from the record of the weather, for one sidereal period of the moon, in order to compare the theory with observation.
[Ill.u.s.tration: Fig. 4]
In the above figure, the circle PER represents the earth, E the equator, PP' the poles, T the centre of the earth, C the mechanical centre of the terral vortex, M the moon, XX' the axis of the vortex, and A the point where the radius vector of the moon pierces the surface of the earth. If we consider the axis of the vortex to be the axis of equilibrium in the system, it is evident that TC will be to CM, as the ma.s.s of the moon to the ma.s.s of the earth. Now, if we take these ma.s.ses respectively as 1 to 72.3, and the moon's mean distance at 238,650 miles, the mean value of TC is equal to this number, divided by the sum of these ma.s.ses,--_i.e._ the mean radius vector of the little orbit, described by the earth's centre around the centre of gravity of the earth and moon, is equal 238650/(72.3+1) = 3,256 miles; and at any other distance of the moon, is equal to that distance, divided by the same sum. Therefore, by taking CT in the inverse ratio of the mean semi-diameter of the moon to the true semi-diameter, we shall have the value of CT at that time. But TA is to TC as radius to the cosine of the arc AR, and RR' are the points on the earth's surface pierced by the axis of the vortex, supposing this axis coincident with the pole of the lunar orbit. If this were so, the calculation would be very short and simple; and it will, perhaps, facilitate the investigation, by considering, for the present, that the two axes do coincide.
In order, also, to simplify the question, we will consider the earth a perfect sphere, having a diameter of 7,900 miles, equal to the actual polar diameter, and therefore TA is equal to 3,950 miles.
In the spherical triangle given on next page, we have given the point A, being the position of the moon in right ascension and declination in the heavens, and considered as terrestrial lat.i.tude and longitude.
Therefore, PA is equal to the complement of the moon's declination, P being the pole of the earth, and L being the pole of the lunar orbit; PL is equal to the obliquity of the lunar orbit, with respect to the earth, and is therefore given by finding the true inclination of the lunar orbit at the time, equal EL, (E being the pole of the ecliptic,) also the true longitude of the ascending node, and the obliquity of the ecliptic PE. Now, as we are supposing the axis of the vortex parallel to the pole of the lunar orbit, and to pierce the earth's surface at R, ARL will evidently all be in the same plane; and, as in the case of A and L, this plane pa.s.ses through the earth's centre, ARL must all lie in the same great circle. Having, therefore, the right ascension of A, and the right ascension of L, we have the angle P. This gives us two sides, and the included angle, to find the side LA. But we have before found the arc AR; we therefore know LR. But in finding LA, we found both the angles L and A, and therefore can find PR, which is equal to the complement of the lat.i.tude sought.
[Ill.u.s.tration: Fig. 5]
We have thus indicated briefly the simple process by which we could find the lat.i.tude of the axis of the central vortex, supposing it to be always coincident with the pole of the lunar orbit. The true problem is more complicated, and the princ.i.p.al modifications, indicated by the theory, are abundantly confirmed by observation. The determination of the inclination of the axis of the vortex, its position in s.p.a.ce at a given time, and the law of its motion, was a work of cheerless labor for a long time. He that has been tantalized by hope for years, and ever on the eve of realization, has found the vision vanish, can understand the feeling which proceeds from frequent disappointment in not finding that, whose existence is almost demonstrated; and more especially when the approximation differs but slightly from the actual phenomena.
The chief difficulty at the outset of these investigations, arose from the conflicting authority of astronomers in relation to the ma.s.s of the moon. We are too apt to confound the precision of the laws of nature, with the perfection of human theories. Man observes the phenomena of the heavens, and derives his means of predicting what will be, from what has been. Hence the motions of the heavenly bodies are known to within a trifling amount of the truth; but it does not follow that the true explanation is always given by theory. If this were so, the ma.s.s of the moon (for instance) ought to be the same, whether deduced from the principle of gravitation or from some other source. This is not so.
Newton found it 1/40 of that of the earth. La Place, from a profound theoretical discussion of the tides, gave it as 1/58.6, while from other sources he found a necessity of diminis.h.i.+ng it still more, to 1/68, and finally as low as 1/75. Bailly, Herschel, and others, from the nutation of the earth's axis, only found 1/80, and the Baron Lindenau deduced the ma.s.s from the same phenomenon 1/88. In a very recent work by Mr. Hind, he uses this last value in certain computations, and remarks, that we shall not be very far wrong in considering it as 1/80 of the ma.s.s of the earth. This shows the uncertainty of the matter in 1852. If astronomy is so perfect as to determine the parallax of a fixed star, which is almost always less than one second, why is it that the ma.s.s of the moon is not more nearly approximated? Every two weeks the sun's longitude is affected by the position of the moon, alternately increasing and diminis.h.i.+ng it, by a quant.i.ty depending solely upon the relative ma.s.s of the earth and moon, and is a gross quant.i.ty compared to the parallax of a star. So, also, the horizontal parallax--the most palpable of all methods--taken by different observers at Berlin, and the Cape of Good Hope, (a very respectable base line, one would suppose,) makes the ma.s.s of the moon greater than its value derived from nutation; the first giving about 1/70, the last about 1/74.2. Does not this declare that it is unsafe to depend too absolutely on the strict wording of the Newtonian law of gravitation. Happily our theory furnishes us with the correct value of the moon's ma.s.s, written legibly on the surface of the earth; and it comes out nearly what these two phenomena always gave it, viz.: 1/72.3 of that of the earth. In another place we shall inquire into the cause of the discrepancy as given by the nutation of the earth.
MOTION OF THE AXIS OF THE VORTEX.
If the axis of the terral vortex does not coincide with the axis of the lunar orbit, we must derive this position from observation, which can only be done by long and careful attention. This difficulty is increased by the uncertainty about the ma.s.s of the moon, already alluded to, and by the fact that there are three vortices in each hemisphere which pa.s.s over _twice_ in each month, and it is not _always_ possible to decide by observation, whether a vortex is ascending or descending, or even to discriminate between them, so as to be a.s.sured that this is the central descending, and that the outer vortex ascending. A better acquaintance, however, with the phenomenon, at last dissipates this uncertainty, and the vortices are then found to pursue their course with that regularity which varies only according to law. The position of the vortex (the central vortex is the one under consideration) then depends on the inclination of its axis to the axis of the earth, and the right ascension of that axis at the given time. For we shall see that an a.s.sumed immobility of the axis of the vortex, would be in direct collision with the principles of the theory.
Let the following figure represent a globe of wood of uniform density throughout. Let this globe be rotated round the axis. It is evident that no change of position of the axis would be produced by the rotation. If we add two equal ma.s.ses of lead at m and m', on opposite sides of the axis, the globe is still in equilibrium, as far as gravity is concerned, and if perfectly spherical and h.o.m.ogeneous it might be suspended from its centre in any position, or a.s.sume indifferently any position in a vessel of water. If, however, the globe is now put into a state of rapid rotation round the axis, and then allowed to float freely in the water, we perceive that it is no longer in a state of equilibrium. The ma.s.s m being more dense than its antagonist particle at n, and having equal velocity, its momentum is greater, and it now tends continually to pull the pole from its perpendicular, without affecting the position of the centre. The same effect is produced by m', and consequently the axis describes the surface of a double cone, whose vertices are at the centre of the globe. If these ma.s.ses of lead had been placed at opposite sides of the axis on the _equator_ of the globe, no such motion would be produced; for we are supposing the globe formed of a hard and unyielding material. In the case of the ethereal vortex of the earth, we must remember there are two different kinds of matter,--one ponderable, the other not ponderable; yet both subject to the same dynamical laws. If we consider the axis of the terral vortex to coincide with the axis of the lunar orbit, the moon and earth are placed in the equatorial plane of the vortex, and consequently there can be no derangement of the equilibrium of the vortex by its own rotation. But even in this case, seeing that the moon's...o...b..t is inclined to the ecliptic, the gravitating power of the sun is exerted on the moon, and of necessity she must quit the equatorial plane of the vortex; for the sun can exert no influence on the _matter_ of the vortex by his attracting power. The moment, however, the moon has left the equatorial plane of the vortex, the principle of momentum comes into play, and a conical motion of the axis of the vortex is produced, by its seeking to follow the moon in her monthly revolution. This case is, however, very different to the ill.u.s.tration we gave. The vortex is a fluid, through which the moon freely wends her way, pa.s.sing through the equatorial plane of the vortex twice in each revolution. These points const.i.tute the moon's nodes on the plane of the vortex, and, from the principles laid down, the force of the moon to disturb the equilibrium of the axis of the vortex, vanishes at these points, and attains a maximum 90 from them. And the effect produced, in pa.s.sing from her ascending to her descending node, is equal and contrary to the effect produced in pa.s.sing from her descending to her ascending node,--reckoning these points on the plane of the vortex.
[Ill.u.s.tration: Fig. 6]
INCLINATION OF THE AXIS.
By whatever means the two planes first became permanently inclined, we see that it is a necessary consequence of the admission of these principles, not only that the axis of the vortex should be drawn aside by the momentum of the earth and moon, ever striving, as it were, to maintain a dynamical balance in the system, in accordance with the simple laws of motion, and ever disturbed by the action of gravitation exerted on the grosser matter of the system; but also, that this axis should follow, the axis of the lunar orbit, at the same mean inclination, during the complete revolution of the node. The mean inclination of the two axes, determined by observation, is 2 45', and the monthly equation, at a maximum, is about 15', being a plus correction in the northern hemisphere, where the moon is between her descending and ascending node, reckoned on the plane of the vortex, and a minus correction, when between her ascending and descending node. And the mean longitude of the node will be the same as the true longitude of the moon's...o...b..t node,--the maximum correction for the true longitude being only about 5 .
[Ill.u.s.tration: Fig. 7]
In the following figure, P is the pole of the earth; E the pole of the ecliptic; L the pole of the lunar orbit; V the mean position of the pole of the vortex at the time; the angle ?EL the true longitude of the pole of the lunar orbit, equal to the _true_ longitude of the ascending node 90. VL is therefore the mean inclination 2 45'; and the little circle, the orbit described by the pole of the vortex _twice_ in each sidereal revolution of the moon. The distance of the pole of the vortex from the mean position V, may be approximately estimated, by multiplying the maximum value 15' by the sine of twice the moon's distance from the node of the vortex, or from its mean position, viz.: the true longitude of the ascending node of the moon on the ecliptic. From this we may calculate the true place of the node, the true obliquity, and the true inclination to the lunar orbit. Having indicated the necessity for this correction, and its numerical coefficient, we shall no longer embarra.s.s the computation by such minutiae, but consider the mean inclination as the true inclination, and the mean place of the node as the true place of the node, and coincident with the ascending node of the moon's...o...b..t on the ecliptic.
POSITION OF THE AXIS OF THE VORTEX.
It is now necessary to prove that the axis of the vortex will still pa.s.s through the centre of gravity of the earth and moon.
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