Rodolfo Amedeo Lanciani.

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containing in addition to the above metals Cu and Cd, follow
the same rula A similar regularitv connects the anhydrous
sulphates of most of these metals, feut an equally remarkable
series is formed by the chlorids of the same metals, which, of
the formula MCI2, have, as far as they have been studied,
atomic volumes which are very nearly equal If we present
these chlorids in tabular form, taking the average of their ob-
served atomic volumes as the real value, we shall find that the
variations from their average are wholly within the limits of
experimental error.

FeCla. Sp. gr. 2-628, Filhol At. vol, found, 60-2'

NiCla •' 2-660, Schiff. '» '* 60-'?

CoCI, " 2-937, P. A J. " " 44-2

CuCla *- 3-054, *' " " 44-0

ZnCla " 2-763, Bodeker. " " 49-4 J- Mean, 47*6.

PtCla •• 6-8696. *» *• " 467 '

MgQa " 2-177, P. &J. " »' 436

CdOl, " 3-6254, Bodeker, '» " 50-5

HgCl, " 6-4032, Kapsten, " *• 60-1^

Now, if we take into account the difficulty of obtaining per-
fectly accurate determinations of for some of these sub-
stances ; and also bear in mind that for most of them only sin-

Sp. gr. calc., 2-668

*» " 2-; 26

" ♦» 2-726

" *• 2-J<25

" u 2'Bb1

*» » 5-638

" •» 1-996

'* '» 3-844

** •• 6-693

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180 F. W, Clarke on Ihe Atomic Volume of Oompoufub.

gle observations have been made, it is clear that every variation
from equality may be ascribed to errors in experiment The
extreme difference in atomic volume is jfrom a minimum of 43*6
to a maximum of 50*7. It is not rare for an equal divergence
to occur between different determinations for a single substence.
A^ain, although chlorids, bromids, and iodids have unequal
atomic volumes, those of similar chlorates, bromates, and
iodates are equal, — at least as far as we have any data.

NaClOa, sp. gr., 2-289, Bodeker. NaBrOj, 3-889, and NaIO„
4-277, Kremers. At vols., 46*5, 45-2, and 46-8.

KClOa, sp. gr., 2-825, Buignet KBrOg, 8-271, and KIO„ 8*979,
Kremers. At vols., 52-7, 51-1, and 58-8.

Most similar compounds of silver and sodium have equal
atomic volumes. This holds true of the chlorids, bromids, and
iodids (as will be shown in another connection hereafter), and
of the sulphates, chlorates, nitrates, and probably carbonates.
The sulphids and oxyds are exceptions.

Some compounds of As, Sb, and Bi, have equal values.
AsjO,, sp. gr., 3-695, Guibourt ; 8-884, FilhoL Sp,Oa, 5-11, Terreil ;
5-78, BouUay. BijOs, 8*079, P. & J. ; 8450, Leroyer k
Dumaa At vols., 51-0—53-6; 50-5—57-1; and 55-4—
In the carbon group compare certain oxyds.
SiOj, 2-668, Devilla TiOj (artificial anatase) 3-700, Hautefeu-
illa SnOj 6-720, Daubr6a At vols., 22-5, 22*2, and 22-3.

Would space permit, I miffht go on multiplying examples to
an almost indefinite extent ; out my object at present is merely
to illustrate certain principles. There are other regularities yet
to be noticed. The first of these is comparatively unimportant,
and I will merely state it as it is, without adducing any evi-
denca All sulphates have atomic volumes which are lower than
the sum of the metallic oxyds and the SO,. That is, the value
for MnS04 is less than the sum of those of MnO and SO^ To
this rule I have found no exceptions. Whether a similar rule
holds good for chromates, moly odates, and tungstates, I am not
certain ; the materials at my command being too limited to set-
tle the question.

Another regularity is more remarkable. It is noteworthy
that certain oxyds and sulphids have atomic volumes which
are lower than those of the metals contained in them. Now, in
a number of cases, I find that those atomic volumes equal ha^
the sum of those of the metal and the other element, provided
that S=15-6, and O receives its highest value, 10*4 I give oxy-
gen this value, for this reason. Many sulphids have atomic
volumes, as I have already stated, equal to the sum of those of

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F. W. Clarke on ike Atomic Volume of (hmpownds. 181

the metal and snlpliur in the free state; so that it is prob-
able that some oxyds follow the same rula 10*4 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: NajS
and K^, whose sp. gr., given by Filhol, are respectivelv 2-471
and 2180. At vols., 81-6 and 51-6. Now 2(22'8)-H5'6 ^ gQ.g .

and -^ ^ ^=58*4. If these theoretical atomic volumes

are true, then the sp. gr. of Na^ will be 2*549, or 0'078 greater
than the value found ; while that of KgS will be 2-059, varying
0-071 from FilhoFs number. These variations are wholly with-
in the limits of error for such compounda

But the results obtained with four oxyds are more striking.

MgO.,KarBten. At vol 12-6. ^55+19^=1 2-1. Error 0*4
CaO. « 3-161, " " 17-7. ^5^=18-1. « 0-4

SrO. " 4-611, FilhoL « 22-4. ?*:^-^*=22-35. " 0-05

AlgO,. " 3-928, Ebelmen. " 26-9. *<^y<"^*> =25-8.^^ 0-1

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 fonnea by the union of these elements
wifi have atomic volumes which are also multiplea Now, sul-
phuric anhydrid, S0», has, according to Buff, the sp. gr. 1*909
at 25°, and according to Baumgartner 1*97^. Its atomic vol-
ume, then, is from 40*5 to 41 '9. And 41*6 is precisely four
times 10*4 1

The chlorids, bromids, and iodids of the alkali metals and
silver, seem to afford a similar exampla In one of my pre-
vious papers I showed that Kopp's values for 01, Br, and 1^ in
their liouid compounds were almost exact multiples of his num-
ber for H, 5-5 Consequently, judging from analogy, it is likely
that these elements in their solid compounds would follow a
similar rule. Now the metals Li, Ka, K, and Ag, have
atomic volumes which do not vary greatly from multiples of
5*5. 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 5*5, that the cir
cumstance cannot be ascribed to accident

I present a tabular view of this regularity.

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1-998, Kremers.

At vol

L fouD

d, 21-3. Calc

L, 22-0.


2*146, Buignet


27-2 "





1-945, Kopp.

• t

38-3 "





6-130, Herapath.


27-9 "





3-079, Kremers.


33-4 "





2-672, P. * J.


44-5 "







" 32-3-32-4 "





3-460, Filhol.


43-6 •*





3-066, Pilhol.


64-3 "


i< '



5-360, Schiff.



43-9 "





182 F. W. Clarke on the Atomic Volume of Compounds,

LiCJL 8p. gr.

NaCl **

KCl "

AgCl "

NaBr "

KBr "

AgBr "

Nal "

KI "

Agl *'

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 corre^nding
compounds of Na ana Ag have nearly equal, or equal atomic
volumes, that the values of potassium compounds exceed those
of the sodium compounds by 1 1*0, (5*5x2), that bromids exceed
chlorids by 5*5, and that iodids have atomic volumes llO
greater than bromids. These regularities in diJBference between
chlorids, bromids, and iodids, however, do not appear so dis-
tinctly 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 wnose atomic volumes are unequal, but sim-
ply related to one another, often form similar compounds hav-
ing ecjual 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

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-

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 1
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 6r that element, or of the lowest value in the group to
which it belongs ? Although this theory cannot be regarded as
entirely proveo, it certainly possesses a considerable deffree of
probability, and seems to harmonize well with the regularities
which I have pointed out But why an element should have a

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S, Newcomb on the apparent inequ/ilities^ etc, 188

higher value in one compound than in another, remains' to be
accounted for, although upon this point, perhaps, Buffs idea
that the different degrees of quanti valence of an element in its
various compounds, cause the differences in its atomic volumes,
mav prove correct*

6ut 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 irutqualities of long
period in the mean motion of the Moon ; by SiMON Newcomb.

[Bead to the Kationnl Academy, April, I8t0.]

The problem of determining the motion of the moon around
the eartn 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 geo-
meter since Newton has added something to the simplicity or
the accuracv of the solution, and, in our own day we nave seen
it successfully completed in its simplest form, in which the earth,
the moon, and the sun are considered as material points, mov-
ing under the influence of their mutual attractions. The satis-
factory solutions are due to the genius of Hansen and of De-
launay. 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 av-
erage discrepancy of less than a second of arc. No doubt could
remain of the substantial correctness of eacL

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 mcreas-
ing, firom century to century, since the time of the earliest re-
corded observations. If we inauire into the problem of these
inequalities of long period, we snail find it seemingly no nearer
a final solution than it was left by La Place, observation
having since added more anomalies than theory has satisfacto-
rily 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 modem ob-

* See Biiffs paper In the Ajmalen d. Ohem. u. Phann., 4th supplement vol,
1866-6. Or, Bee Ids ** Q-nmdlehren der theoretisohen Ohemte.*'

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184 S, Newcomb on Ae apparent mequalUies

servationi Its cause was first discovered hy La Place, who
showed that it was due to the efiect of the action of the planets
in changing the eccentricity of the earth's orbit

The results of his computations agreed substantially with ob-
servations, 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 dif-
ference is now accounted for by an increase in the length of the
day. That this length is increasing is also known firom theoret-
ical considerations, but the data mr its accurate determination
are wanting.*

In the third volume of the Mecanique CSJeste (Seconde Partie,
Livre vii, Chapitre v) La Place discusses an apparent inequal-
ity of long period in the motion of the moon. The discussion
is mainly empirical The existence of the inequality is inferred
fix>m observations, these showing that the mean motion of the
moon during the half century following 1756 was less than dur-
ing the half century preceding. He then assumed that the in-
equality was due to tne 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 d^rees
per annum, and determined the coefficient of the varying angle
solely from the observationa The result was that these might
be satisfied by supposing the inequality of mean longitude

«=:4r-61 [or 16"-39] sin (2 il^+n^^SnQ)

t£, in this expression, we substitute Hansen's values of the
elements, it becomes

«=:16"-39 sin [178^ 26'+(l^ 57'-4) (^-1800)].
When in 1811 Burckhardt constructed his tables of the moon,

* The time and place when the diaoordanoe referred to was first distinctly attrib-
uted to the tidal retardation of the earth having been a subject of discussion, tiie
following extract from an article on " Modem Theoretical Astronomy** in the North
American Review for October, 1861 (voL 93| p. 886), may not be devoid of interest

" It seems to be well established that the new theory is inconsistent with the ob-
servations of ancient edipses, and if it should prove to be correct we may be driven
to the conclusion, that a portion of the acceleration proceeds firom some other cause
than the attraction of gravitation, or that tiie lengtii of the day is gradually increas-
ing to an eztcDt which has become perceptible fh>m tiie cause to which we have
already referred [the tidal retardation, p. 374]. If, as centuries roll by, the day
should graduaJly mcrease, the moon would move a little farther in the course of a
day than if no such increase should take place. Shice, in our calculatiaw, we sup-
pose the day constant, the apparent acceleration would be greater than the real —
precisely the effect observed. The difference can be entirely accounted for by sup-
posing an increase of something leas than one thousandth of a second per ceotuiy
m the length of the day, and a oorreipowling dimiaiitioii in tiie lunar month.*'

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of long period in the ifiean motion of the Moon, 185

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

W=-r2"-6 cos [291° 5r+(2^ 0'-46)(^-1800y]
= 12"-6 sin [201° 57'+(2° 0'-45)(«— 1800)]

Succeeding investigators have regarded the theoretical coeffi-
cients of botn of these terms as insensible. It does not seem
likely that there is any such difference between the two terres-
trial 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, 3D - 2F - Z+ Sf;

In Hansen's, w—Sw'.

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 (juestion. 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 coeffi-
cients, and says he has not been able to find any sensible coeffi-
cient for La rlace's argument of long period. But on examin-
ing 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:

81= 16"-0l sin (-^— 16^'+18.^"+36*' 20').
g, g' and g" being the mean anomilies of the moon, the earth
and Venus respectively. As this expression still failed to ac-
count for the observed variations of the moon's lonmtude he
continued the approximation to the fourth power of the dis-
turbing force, and found that the terms of the third and fourth
order increased the coefficient to 27'' 4, the angle remaining un-
changed, so that the term became

2r'-4 sin (-*.^-.16^+18^'+86° 20'),
But this increase made the theory rather worse, and the term
depending on the argument of Ajry's equation between the
earth and Venus was then tried with the result —
dl z= 23"-2 sm (8^"- 13^'+ 316° 30').
The introduction of this term seemed to reconcile the theory
with observation.

Am. Jour. Sol— Sbooud Sbbzbs, You L, No. 149.— Ssft., 1870«

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186 S. Newcomb on the apparent inequalities

Hansen finally remarks that these values of the coefficients
are still subject to some uncertainty fi*om his not having em-
ployed decimals enough in his computation.

fn a letter to the Astronomer Boval, published in the Monthly
Notices of the Royal Astronomicfd Society for Nov. 1854, Han-
sen gives a statement of the elements employed in his tables of
the moon, and refers to the subject of these inequalities in the
following terms : —

"The accurate determination of these two inecjualities by
theory is the most difficult matter which presents itself in the
theory of the moon's motion. I have on two occasions and by
different methods sought to determine their values, but I have
obtained results essentially different fix>m each other. I am
now again engaged with their theoretical determination by a
method which I have simplified, and hope to bring the opera-
tion to a definitive close. I have also applied to my tables
some coefficients which are not fi'ee from empiricism but which
I can justify by the circumstance that they represent the ancient
as well as tne modem observations with great exactness, and it
may be expected that they will represent the future observa-
tions equally well."

Hansen's lunar tables were published in 1857.

The terms of long period finally adopted are

16"-34 sin (-.^—leE-hlSV+SO^ 12')
+21-48 sin (8V-13E+274° 14'),

V and E representing the mean longitudes of Venus and the
earth. Changing them to mean anomalies the terms become
16"-34 sin (-^-16flr'+l8^'+33° 36')
+21-47 sin (8^"— 13/+4° 44').

It appears that while the first term has been restored to what
was substantially its original value, when only the first power
of the disturbing force was included, the argument of the second
term has been cnanged by 50^, the coefficient being but slight-
ly changed.

In a letter to the Astronomer Royal, dated 1861, Feb. 2d,
found in the Monthly Notices for March, 1861, Hansen a^in
refers to this second term with the statement that its coefficient
is one of those somewhat empirical At the same time he has
found the coefficient, by his last theoretical determination of it,
by no means insensible, like Delaunay. He adds that in the
comparison with observation he has never gone beyond Brad-
ley, nevertheless his tables satisfactorily represent the ancient

A well marked feature of Hansen's published works is the co-
piousness and perspecuity with which his theoretical calcula-
tions are laid dovm. But, so fer as I am aware he has never
published any computation of these inequalities except that
part of the first inequality which depends on the first power of

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of long period in the mean motion of the Moon. 187

the disturbing force. This computation is found in voL xvi of
the Memoirs of the Eoyal Astronomical Society. In the sec-
ond part of his " Darlegung " we find a general method of treat-
ing inequalities of long period, but — unless I have overlooked
it — ^no computation of any particular inequali^. Nor do we
find any statements of the numerical results of Hansen's various
computations except those alreadv quoted.

Tne onlv geometer besides Hansen who has attacked the
problem of these inequalities is Delaunay. His researches are

?ublished in full in the Additions to the Connaissance des
'emps for years 1862 and 1863. For the first approximation
to the first inequality his result is

16"-02 sin (-/-16r+18r+35° 20'-2)
a result practicallv identical with that of Hansen. The ulterior
approximations cfiange it to

16''-84 sm (-/— l6r+18r+35° 16'-5),
80 that they increase the coefficient instead of diminishinff it as
in Hansen's theory. The difference is however so small that
the results may be regarded as identical

But, in the case of the second inequality instead of reproduc-
ing the result of Hansen, he finds a coefficient of only 0'''27, a
quantity quite insignificant in the present state of the question.
We have thus an irreconcilable difference on a purely theoreti-
cal question.

I propose to inquire whether we have in either theory an en-
tirely satisfactory agreement with observation. As a prelimin-
ary step to this inquiry I have prepared the following table of
the mean longitude of the moon frdm the tables of Burckhardt
and of Hansen respectively, for a series of equidistant dates, the
interval being 8662*5 days, and the epoch 1800 Jan. 0, Greenwich
mean noon. These dates are marked by the year near the
b^inning of which they falL Colunm L^ gives Burckhardt's
mean longitude on the supposition of uniform motion, fi:om the
data given on the fifth page of the introduction to his tablea
Next is given the acceleration of the mean longitude deduced
from Table xlviii. The inequality of long period is from Table
XLIX. The sum of these three quantities gives the corrected
mean longituda

Hansen's mean longitude and secular acceleration are deduced
in the same way from the elements given on page 15 of his
Tables de la Lune. His terms of long period are deduced from
Tables XLi and xui, the constants bein^ subtracted and the re-
mainder reduced to arc by being multiplied bv the factor
0"-004703. The last column of the table gives the correction
to Burckhardt's mean longitude to reduce it to that of Hansen.
That this difference is really the mean difference between the
longitudes of the moon deduced from the two tables is shown

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S. Newcomb on the apparent inequaUtiea

by its agreement with the known difference at particular epochs.
At the end of the British Nautical Almanac for 1862 is found a
comparison of the two tables, from which it appears that Burck-
hardt's mean longitude was then greater than Hansen's by about
14''-2. The general agreement oetween 1750 and 1800, when
both tables agreed with observations, shows that the difference
of mean motion is certainly affected with no sensible error.








Corr. Mmo






/ //
100 19 28-0

+ 4-9

- U

100 19 24-9

/ // //

18 14-4 +38-6


' //
100 18 31-5



347 6 46-4

+ 3-6


347 6 38-2

6 3«>-8 +34-1

Online LibraryRodolfo Amedeo LancianiThe American journal of science and arts → online text (page 74 of 109)