the metal and sulphur in the free state; so that it is probable that some oxyds follow the same rule. 104 being the highest number for oxygen, renders it likely that that is the true value for this element in those oxyds corresponding to the above mentioned sulphids.

Now, two sulphids appear to follow this rule, viz: Na2S and KS, whose sp. gr., given by Filhol, are respectively 2:471 and 2.130. At. vols., 31.6 and 51.6. Now 2(22.8)+15.6

and

2(45.6)+15.6.
2

2

=

30.6;

=534. If these theoretical atomic volumes

are true, then the sp. gr. of NagS will be 2.549, or 0.078 greater than the value found; while that of KS will be 2.059, varying 0-071 from Filhol's number. These variations are wholly within the limits of error for such compounds.

But the results obtained with four oxyds are more striking. MgO. Sp. gr. 3.200, Karsten. At. vol. 12.5. 138+104-12.1. Error 0.4

2

The oxyds of barium, sodium, and potassium, do not follow this rule.

One more regularity traced, and I am done. If my views. concerning multiple relations are correct, and all the values for oxygen and sulphur are multiples of the lowest, then we must expect that compounds formed by the union of these elements will have atomic volumes which are also multiples. Now, sulphuric anhydrid, SO,, has, according to Buff, the sp. gr. 1-909 at 25°, and according to Baumgartner 1.975. Its atomic volume, then, is from 405 to 41.9. And 416 is precisely four times 10.4!

The chlorids, bromids, and iodids of the alkali metals and silver, seem to afford a similar example. In one of my previous papers I showed that Kopp's values for Cl, Br, and I, in their liquid compounds were almost exact multiples of his number for H, 55 Consequently, judging from analogy, it is likely that these elements in their solid compounds would follow a similar rule. Now the metals Li, Na, K, and Ag, have atomic volumes which do not vary greatly from multiples of 55. And, in accordance with what we should expect, their chlorids, bromids, and iodids have atomic volumes which are either exact, or so nearly exact, multiples of 55, that the cir cumstance cannot be ascribed to accident.

I present a tabular view of this regularity.

LiCl. Sp. gr. 1998, Kremers. At. vol. found, 21.3. Calc., 22-0. Sp. gr. calc., 1932

On comparing these numbers, it will at once be noticed, not only that the variations from theory are remarkably slight, but also (which has been previously stated) that the corresponding compounds of Na and Ag have nearly equal, or equal atomic volumes, that the values of potassium compounds exceed those of the sodium compounds by 110, (5.5×2), that bromids exceed chlorids by 55, and that iodids have atomic volumes 110 greater than bromids. These regularities in difference between chlorids, bromids, and iodids, however, do not appear so distinctly in other series of them.

Now, to sum up the important relations traced in this paper, bearing in mind that in many cases exceptions exist.

First. When similar metals have equal atomic volumes, those of their similar compounds will also be equal.

Second. Metals whose atomic volumes are unequal, but simply related to one another, often form similar compounds having equal values.

Third. Some compounds have atomic volumes which stand in very simple relations to the sums of those of the free elements which form them.

Fourth. Compounds formed by the union of similar elements, have atomic volumes which are multiples of the lowest for that group.

And, fifth, we may add the multiple relations traced in my former papers, which not only connect the atomic volumes of different elements, but the various values for each single ele

ment.

Now, what do these regularities mean. Are we justified either in drawing any direct conclusions from them, or in basing upon them any general theory of atomic volumes? To this must answer, that, although no generalization is absolutely established by them, it seems to me that one is decidedly hinted at. May we not say that in all compounds, the atomic volume of every element will be either a perfect multiple of the lowest value for that element, or of the lowest value in the group to which it belongs? Although this theory cannot be regarded as entirely proved, it certainly possesses a considerable degree of probability, and seems to harmonize well with the regularities which I have pointed out. But why an element should have a

higher value in one compound than in another, remains to be accounted for, although upon this point, perhaps, Buff's idea that the different degrees of quantivalence of an element in its various compounds, cause the differences in its atomic volumes, may prove correct.

*

But at all events, whether the theory which I put forward turns out true or false, it may, perhaps, by lending some system to the study of atomic volume, pave the way for something of greater value.

Boston, May 30th, 1870.

ART. XIX. Considerations on the apparent inequalities of long period in the mean motion of the Moon; by SIMON NEWCOMB.

[Read to the National Academy, April, 1870.]

THE problem of determining the motion of the moon around the earth under the influence of the combined attraction of the sun and planets has, more than any other, called forth the efforts of mathematicians and astronomers. Nearly every great geometer since Newton has added something to the simplicity or the accuracy of the solution, and, in our own day we have seen it successfully completed in its simplest form, in which the earth, the moon, and the sun are considered as material points, moving under the influence of their mutual attractions. The satisfactory solutions are due to the genius of Hansen and of Delaunay. Working independently of each other, each using a method of his own invention more rigorous than had before been applied, they arrived at expressions for the longitude of the moon which, being compared, were found to exhibit an average discrepancy of less than a second of arc. No doubt could remain of the substantial correctness of each.

The solutions here referred to exhibit only inequalities of short period in the motion of the moon. But, it has long been known, from observation, that the mean motion of the moon is subject to apparent changes of very long period, and especially to a secular acceleration by which it has been gradually increasing, from century to century, since the time of the earliest recorded observations. If we inquire into the problem of these inequalities of long period, we shall find it seemingly no nearer a final solution than it was left by La Place, observation having since added more anomalies than theory has satisfactorily shown to exist.

The first inequality in the order of discovery was the secular acceleration. This was discovered about the middle of the last century by a comparison of ancient eclipses with modern ob

*See Buff's paper in the Annalen d. Chem. u. Pharm., 4th supplement vol., 1865-6. Or, see his "Grundlehren der theoretischen Chemie."

servations. Its cause was first discovered by La Place, who showed that it was due to the effect of the action of the planets in changing the eccentricity of the earth's orbit.

The results of his computations agreed substantially with observations, and was therefore received with entire confidence until less than twenty years ago. The question being then taken up by Mr. John C. Adams, this eminent mathematician was led to the conclusion that La Place's result was nearly twice too large.

The same conclusion was reached independently by Delaunay, and gave rise to a remarkable discussion, the history of which is too familiar to be now recounted. It is now conceded that the value found by Adams and Delaunay is theoretically correct.

The new result no longer agreeing with observation, the difference is now accounted for by an increase in the length of the day. That this length is increasing is also known from theoret ical considerations, but the data for its accurate determination are wanting.*

In the third volume of the Mécanique Céleste (Seconde Partie, Livre vii, Chapitre v) La Place discusses an apparent inequality of long period in the motion of the moon. The discussion

is mainly empirical. The existence of the inequality is inferred from observations, these showing that the mean motion of the moon during the half century following 1756 was less than during the half century preceding. He then assumed that the inequality was due to the fact that twice the mean motion of the moon's node, plus the motion of its perigee, minus that of the sun's perigee was a very small quantity, less than two degrees per annum, and determined the coefficient of the varying angle solely from the observations. The result was that these might be satisfied by supposing the inequality of mean longitude Si 47-51 [or 15"-39] sin (2 DD37)

If, in this expression, we substitute Hansen's values of the elements, it becomes

-15" 39 sin [173° 26'+(1° 57'-4) (t-1800)].

When in 1811 Burckhardt constructed his tables of the moon, * The time and place when the discordance referred to was first distinctly attributed to the tidal retardation of the earth having been a subject of discussion, the following extract from an article on “Modern Theoretical Astronomy" in the North American Review for October, 1861 (vol. 93, p. 385), may not be devoid of interest

It seems to be well established that the new theory is inconsistent with the observations of ancient eclipses, and if it should prove to be correct, we may be driven to the conclusion, that a portion of the acceleration proceeds from some other cause than the attraction of gravitation, or that the length of the day is gradually increasing to an extent which has become perceptible from the cause to which we have already referred [the tidal retardation, p. 374]. If, as centuries roll by, the day should gradually increase, the moon would move a little farther in the course of a day than if no such increase should take place. Since, in our calculations, we sup pose the day constant, the apparent acceleration would be greater than the realprecisely the effect observed. The difference can be entirely accounted for by supposing an increase of something less than one thousandth of a second per century in the length of the day, and a corresponding diminution in the lunar month."

he omitted the sun's perigee from this argument by the authority of La Place, himself, who now attributed the inequality to a difference of compression between the two hemispheres of the earth. The function was also changed from sin to cos and the coefficient altered. The adopted term thus became

81-12"5 cos [291° 57'+(2° 0'45) (t-1800)]

= 12"-5 sin [201° 57'+(2° 0'45) (t-1800)]

Succeeding investigators have regarded the theoretical coefficients of both of these terms as insensible. It does not seem likely that there is any such difference between the two terrestrial hemispheres as could produce the second, but I am not aware that the coefficient of the first has ever been shown to be insensible by any published computation. This coefficient is of the ninth order and the argument is,

In Delaunay's notation,
In Hansen's,

3D-2F-1+31';
w-3w'.

The period is 184 years, and the large value of the ratio of this period to that of the moon itself might render the coefficient sensible. Both Hansen and Delaunay pronounce it insensible, but neither publish their computations of its magnitude.

These terms have ceased to figure in the theory of the moon since Hansen announced that the action of Venus was capable of producing inequalities of the kind in question. So far as I am aware, Hansen's first publication on this subject is that found in No. 597 of the Astronomische Nachrichten (B. 25, S. 325.) Here, in a letter dated March 12, he alludes to La Place's coefficients, and says he has not been able to find any sensible coefficient for La Place's argument of long period. But on examining the action of Venus on the moon he found, considering only the first power of the disturbing force, the following term in the moon's mean longitude:

87 = 16" 01 sin (-g-16g'+18g"+35° 20′).

9, g' and g" being the mean anomilies of the moon, the earth and Venus respectively. As this expression still failed to account for the observed variations of the moon's longitude he continued the approximation to the fourth power of the disturbing force, and found that the terms of the third and fourth order increased the coefficient to 27" 4, the angle remaining unchanged, so that the term became

27" 4 sin (g-16g'+18g"+35° 20′),

But this increase made the theory rather worse, and the term depending on the argument of Airy's equation between the earth and Venus was then tried with the result

123" 2 sin (8g"-13g'+315° 30').

The introduction of this term seemed to reconcile the theory with observation.

AM. JOUR. SCI.-SECOND SERIES, VOL. L, No. 149.-SEPT., 1870,