Authority. MELTING POINTS-continued. Oxides. *Cupric oxide, CuO 1055 (By interpolation).-J.J. Molybdenum trioxide, M003 759 Carnelley. Boron trioxide, B203 577 NOTE ON THE MOON'S VARIATION AND PARALLACTIC INEQUALITY. BY REV. M. H. CLOSE, M.A. [Read JUNE 8, 1891.] VARIOUS writers on celestial dynamics, including some of the first rank, and authors of Treatises, Manuals, &c., of Astronomy, have given an elementary deduction of the Variation inequalities in the moon's motion (without the numerical values) from the action of the solar differential perturbing forces, and also an elementary proof that the shortest axis of the Variation orbit, considered apart from all other inequalities, is directed towards the sun. Newton was the first to do this in the Principia, Bk. I., Prop. 66, cors. 2-5. But he has also given us his masterful geometrical exposition of the subject in the third Book of the Principia, which was considered by Laplace to be one of the most wonderful parts of that wonderful work. The investigation has been since carried out more fully by the more powerful method of analysis. As, then, we are not dependent on the said elementary deduction and proof, we incur no great loss when noting what, apparently, has not yet been pointed out, at least in print, viz., that they are quite insufficient for their intended purpose. They were at first put forward in the light of knowledge, otherwise obtained, of the phenomena in question ; and the known correctness of the conclusions to which they seem to lead has concealed the inconsequentiality of the reasoning, both from the original author and from the others who probably have followed him in this matter, even down to the present day. The accompanying diagrams of the Variation and of the Parallactic Inequality are both given with only such detail as is necessary for the present purpose; the arrows, of course, represent merely the positions and directions, not the magnitudes, of the various forces. First, we have to consider the elementary deduction of the Variation inequalities in the moon's motion from the action of the solar perturbing forces. For simplicity, the moon's undisturbed orbit is supposed to be circular, It is said that, as in the quadrants immediately preceding syzygies the solar tangential disturbing force acts in consequentia, or in the direction of the moon's motion, therefore the moon's velocity is increasing all through those quadrants, reaching its maximum at syzygies ; and for corresponding reasons the velocity is decreasing in the other two quadrants, reaching its minimum at both quadratures. All these statements of fact are, of course, quite right. But the “therefore” is unwarranted; for this argument ignores what, for all we yet know to the contrary, might be a very important element of the question, namely, the evidently necessary deformation of the moon's orbit by the disturbing forces. This deformation would give rise to a tangential component of the earth's attraction on the moon, which we shall call the terrestrial tangential forces; and, so far, we know nothing of the possibilities as to the magnitude, and the effects, of this force. The sun produces the moon's Variation in longitude in two ways, first by the production of said deformation of the orbit by the whole general action of his disturbing forces, both tangential and radial, which deformation originates said terrestrial tangential forces; and secondly, by the immediate local action of his tangential forces in modifying the velocity of the moon in her orbit. For all that we are told in this argument to the contrary, the effect of the former might be greater (which indeed it actually is) than that of the latter, and it might be in the opposite direction; as it would be if the Variation orbit, with its present ovalness, had its longest axis directed to the sun; this being what learners (for whom this argument is intended) would naturally expect. Now it so happens that we have only to take the short step fresm the Variation to the Parallactic Inequality to meet with an obtrusive illustration of the antecedent possibility of what we have contemplated. In the Parallactic Inequality orbit, the moon quickens or slackens her pace always in opposition to the solar tangential forces. The reason of this is that the accumulated deformation of the moon's orbit, supposed to be originally circular and devoid of all other inequalities, by the whole general action of the Parallactic Inequality disturbing forces, makes the sunward radius vector a maximum, and the opposite one a minimum. And the difference, though relatively very (, ассета small, is such that, while the moon is going from opposition to conjunction, the retardation due to her rising against the earth's attraction exceeds the acceleration due to the immediate action of the solar tangential forces; and vice versa, when she is going from conjunction to opposition. So, then, the argument to which we are now demurring would be downright wrong if applied to the Parallactic Inequality orbit; and we are given no reason why it must necessarily apply to the Variation orbit. Let us pause for a moment over the interesting circumstances that the effect of the terrestrial tangential forces on the moon's motion in longitude is greater than the immediate direct effect of the solar ones, both in the Variation and in the Parallactic Inequality. As to the Variation-we can claim for the earth that it is working more than half of this lunar inequality; though we have to admit that it is the sun which has enabled the earth to do this by having produced the Variation deformation of the moon's orbit. In this case the two sets of tangen Fig. 1.-Variation. tial forces fortunately co-operate; so that the whole result is the sum of their separate effects. (By the way, this circumstance has helped to disguise the inconclusiveness of the above argument.) As to the Parallactic Inequality-we can claim for the earth, not only that it is working the whole of this lunar inequality, but that it is, in addition, cancelling the immediate local effect of the solar tangential forces. In this case, the forces being unfortunately in antagonism, the net result is only the FiG. difference of their separate effects. Of course we admit, as before, that it is the sun's own general action in producing the Parallactic Inequality deformation of the moon's orbit which enables the earth to accomplish so much. (See diagram'.) In each diagram the moon's orbit is represented by a continuous line. The solar tangential forces are indicated by the outside arrows drawn with dotted lines, and the terrestrial tangential forces by the inside arrows drawn with broken lines. E is the earth; the sun is to the right. In the second diagram the eccentricity of the earth's position has been magnified 300 times. A FIG. 2.-Parall. I Secondly, we have to consider the elementary proof that the shortest axis of the Variation orbit is directed towards the sun. The argument consists of three links (a), (b), and (c), which are so connected that the insufficiency of any one is enough to vitiate the whole chain of the reasoning. But this is not all; it will be found that, curiously enough, each one of the three is, in itself, incapable of bearing the stress put upon it. We give here all the links of the argument; though we have had the first one already in a different connexion. (a) It is said that, as in the quadrant immediately preceding conjunction the solar tangential force is acting in the direction of the moon's motion, until she arrives at conjunction, where the force changes its direction, therefore the moon's velocity is increasing until conjunction is reached, where it is a maximum. (6) It is then said that, the moon's velocity being, thus, a maximum at conjunction, the Variation orbit must be flattest at that point, supposing other things to be equal. But—it is added—other things are even better than equal; because at that point the sun's radial perturbing force directed away from the earth is a maximum ; and consequently the whole earthward pull on the moon is a minimum. Therefore—it is added—the moon's velocity being greatest at conjunction, and the deflecting earthward pull least, the Variation orbit is, for two reasons, flattest at that point; and for corresponding reasons, mutatis mutandis, it is most curved at quadrature. (©) It is then said, finally, that as the Variation orbit is, thus, flattest at conjunction, and most curved at quadrature, therefore the moon's radius vector must be a minimum at the former, and a maximum at the latter. In reply to (a)— We have seen already that this argument, as it stands, is quite inconclusive. But we may now add that, in its present connexion, it is a veiled, partial petitio principii ; it goes far towards assuming that the longest axis of the Variation orbit is not directed towards the sun; to prove which is its ultimate aim. (It is self-evident that either the longest or the shortest axis must be so directed.) In reply to (6)—This argument contains, very singularly, a statement which is actually contrary to fact. It will be found that the increase of the earth's attraction on the moon at conjunction in consequence of the diminution of the moon's radius vector, exceeds the out |