almost uniformly less than the calculated value. This deficiency of pressure is doubtless to be accounted for by a fact which MM. Troost and Hautefeuille have noticed in this connection. The protochloride of phosphorus deviates quite appreciably from the laws of Mariotte, Gay-Lussac, and Avogadro, the product of the volume and pressure of a given quantity of vapor at 180° and the pressure of one atmosphere being 1-548 per cent less than at the same temperature and the pressure of one-half an atmosphere.* Now we may assume as a general rule that when the product of volume and pressure of a gas is slightly less than_the_theoretical number (calculated by the laws of Mariotte, Gay-Lussac, and Avogadro) the difference for any same temperature is nearly proportional to the pressure.† It is therefore probable that between 160° and 180°, at pressures of about one atmosphere, the product of volume and pressure for protochloride of phosphorus is somewhat more than three per cent less than the theoretical number. The experiments of Wurtz, as exhibited in Table IX, show that the pressure, and therefore the product of volume and pressure, (we may evidently give the volume any constant value as unity,) in a mixture consisting principally of the protochloride is on the average a little more than two per cent less than is demanded by theory, the differences being greater when the proportion of the protochloride is greater. The deviation from the calculated values is therefore in the same direction and about such in quantity as we should expect.‡

M. Wurtz has remarked that the average value of (the density of the possible perchloride) is nearly identical with the theoretical density of the perchloride, and appears inclined to attribute the variations from this value to the errors of experiment. Yet it appears very distinctly in Table IX, in which the experiments are arranged according to the value of (the pressure due to the possible perchloride), that increases as π diminishes. The experiments of MM. Troost and Hautefeuille show that the coincidence remarked by M. Wurtz is due to the fact that on the average in these experiments the deficiency of the density of the possible perchloride (compared with the *Troost and Hautefeuille, Comptes Rendus, vol. lxxxiii (1876), p. 334. Andrews, "On the Gaseous State of Matter." Phil. Trans., vol. clxvi (1876),

p. 447.

The deviation of the protochloride of phosphorus from the laws of ideal gases shows the imposibility of any very close agreement between such equations as have been deduced in this paper and the results of experiment in the case of gasmixtures in which this substance is one of the components. With respect to the question whether future experiments on the vapor of the perchloride (alone, or with an excess of chlorine or of the protochloride), will reduce the disagreement between the calculated and observed values to such magnitudes as occur in the case of the protochloride alone, it would be rash to attempt to anticipate the result of experiment.

theoretical value) is counterbalanced by the excess of density of the protochloride. When > 400, the effect of the deficiency in the density of the possible perchloride distinctly preponderates; when <250, the effect of the excess of density in the protochloride distinctly preponderates. But the magnitude of the differences concerned is not such as to invalidate the general conclusion established by the experiments of M. Wurtz, that the dissociation of the perchloride may be prevented (at least approximately) by mixing it with a large quantity of the protochloride.

Table for facilitating calculation.-The numerical solution of equations (10), (11), (12) and (13) for given values of t and p may be facilitated by the use of a table. If we set

By these equations the values of L are easily calculated. The values of may then be obtained by inspection (with interpolation when necessary) of the following table. From 4 the value of D may be obtained by multiplying by D,, viz., by 1.589 for peroxide of nitrogen or formic acid, by 2-073 for acetic acid, and by 3.6 for perchloride of phosphorus.*

The constants of these equations are of course subject to correction by future experiments, which must also decide the more general question-in what cases, and within what limits,

*The value of diminished by unity expresses the ratio of the number of the molecules of the more complex type to the whole number of molecules. Thus, if A=120, in the case of peroxide of nitrogen there are 20 molecules of the type N20 to 80 of the type NO,, or in the case of perchloride of phosphorus there are 20 molecules of the type PC1, to 40 of the type PC1, and 40 of the type C.. A consideration of the varying values of A is therefore more instructive than that of the values of D, and it would in some respects be better to make the comparison of theory and experiment with respect to the values of A.

and with what degree of approximation, the actual relations can be expressed by equations of such form. In the case of perchloride of phosphorus especially, the formula proposed requires confirmation.

ART. XLIV.-On a secular inequality in the Moon's Motion produced by the oblateness of the Earth; by J. N. STOCKWELL.

HAVING been engaged, during a number of years past, in a thorough and systematic examination of the physical theory of the moon's motion, it seems proper to make known to astronomers, in advance of the publication of my researches which are now essentially completed, one of the most curious and interesting results at which I have arrived relative to the motion of our satellite.

It has been known, since the time of Newton, that the attraction of a spheroidal body on a point without its surface is different from that of a sphere having the same mass. If the spheroid be one of revolution, like the earth, the attraction depends not only on the distance of the attracted point from

AM. JOUR. SCI.-THIRD SERIES, VOL. XVIII.-No. 107, Nov., 1879.

the earth's center, but also on its distance from the equator. If the attracted point were situated in the plane of the earth's equator the attraction of the earth upon it would be greater at a given distance than if the earth were spherical. The attraction would also be greater either north or south of the equator until we reached the paralled of about 35° 16', at which points the attraction of the earth would be nearly independent of its spheroidal form. For all points situated beyond the parallels of 35° 16' the attraction of the earth is less than it would be if it were spherical.

From these general considerations we may draw the following conclusions: First. A body would revolve round the earth, at a given distance from its center, in less time if it moved in the plane of the equator, than it would if the earth were spherical; and its motion would be uniform. Second. The time of revolution would be increased if the body moved in a plane inclined to the equator, and its motion would not be uniform, on account of the redundancy or deficiency of matter beneath the different parts of its course. It is evident that the

motion in an orbit perpendicular to the equator would suffer greater variations from the unequal distribution of matter, than it would for any other inclination.

We shall now apply the preceding considerations to the motion of the moon around the earth, supposing for greater simplicity that her orbit is circular.

Since the inclination of the moon's orbit to the equator is always less than 35° 16', it follows that the earth's attraction on the moon is always greater than it would be if the earth were spherical. But since the inclination varies between the limits of about 18° 19' and 28° 35' during a period of about nineteen years, it follows that the earth's attraction undergoes sensible variations; and hence the moon's place at any given time requires to be corrected on account of the varying inclination of its orbit to the equator. The corrections to the moon's longitude and latitude arising from this cause have been calculated, and applied to the moon's place during the whole of the present century. All these varying inequalities in the forces would accurately compensate each other during each revolution of the moon's node, provided the mean inclination of the lunar orbit to the equator always retained the same value. Now the mean inclination of the moon's orbit to the equator is the same as the inclination of the ecliptic to the same plane; and since the inclination of the ecliptic to the equator is slowly becoming less, it follows that the plane of the moon's orbit is gradually approaching the plane of the equator; and hence its mean motion must be increasing. All these various conclusions are fully confirmed by mathematical analysis, and were first suggested by it.

Having thus shown the existence of a secular inequality in the moon's motion depending on the oblateness of the earth, it only remains to determine its amount. But as a mathematical analysis of the problem is not within the scope of the present paper, I shall be content with a mere statement of the semigeneral formula together with its numerical value.

If we put &, for the obliquity of the ecliptic in 1850, and e for its value at any time t, and also suppose that the ellipticity of the earth is, I find the following value for the secular inequality depending on the earth's oblateness, namely:

Sv=+24"-827 (sin'ε-sin'ε)dt.

If we develop the integral into a series and retain only the first term we shall have

(sin'ε-sin') dt +0.008675 i,

in which i denotes the number of centuries counting from 1850 Hence the secular inequality becomes

Sv=0" 1981 ".

This term, though small, is of sufficient importance to be used in computing ancient eclipses.

In conclusion I would state that I have found several inequalities in the moon's motion which are not recognized by existing theories, of even greater practical interest and importance than the one to which I have called attention in this Cleveland, Oct. 2, 1879.

paper.

ART. XLV. Discovery of two new Asteroids; by Professor C. H. F. PETERS. Communication to the Editors dated Litchfield Observatory of Hamilton College, Clinton, N. Y., October 6, 1879.

Two more planets of the asteroid group were found by me in the month of September, respectively on the 11th and 25th. I communicate the observations hitherto obtained.

The magnitude of the first is now 11m0, that of the latter

10m.5.