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6
THE LIBRARY
F HE UNIVERSITY OF
BRITISH COLUMBIA
J
]
INVESTIGATION
OP
CORRECTIONS TO HANSEN'S TABLES OF THE MOON;
•vT:a n
•4
TAHLKS von TIIKIU vV rPLICATlON.
BV
S T IVt O N iNl E W C < ) M H ,
I'UoKKssoi;, r. s. n.vvv.
FORMING PART III OF PAPERS PURUSIIED BY THE COMMISSION ON THE TRANSIT OF VENUS.
WASHINGTON:
a O V E R N M E N r P 11 I N T I N a O F I'M C
1870.
diiltBCHHSHBaesSS
TABLE OF CONTENTS
'^^l
Vagi'.
INTROIILCTORV NOTE
i
i i.-INVICSncJATION or ERRORS OF LONGITUDE.
Evection 8
Variation 8
Mraii urior iif (alxiiar liylit a>cciisii)n at <liiri'iLnt tiiius of (lay i)
N'aliic of solar paiallax cniploved i)
l.isi of corrci lions to " .\ri;iimtiit Fonilaiiuiilal ' lo
(HMK'ral Ideas which form tlic liasis of this investigation li
Dill'ercntial cofllicients 12
Mean apparent error of Hansen's Tallies in riyht asiension 12
Siiilden apparent alteration in mean motion of the moon 1 1
Corrections for limli and oliservatory lo render oljservalioK;. sirielly eoniparalile 13
Mean ontstaiulinn talinlar error of the moon in lon.nitiule 13
Corrections of sliort perioil acliially applied 14
Ivpiation connectini; eriurs of moon's laludar right ascension with errors of Innar elements
Hnins ol eriors of moon's 1 orr^ried right ascensitiii given In ohservalioiis at (ireenwich .ind Washington. ... 17
Normal ei|ualions foi deteriiiiTiing (1/, «, and /(■ liy least sipiares 20
Values of outstanding errois of lunar elements for each year 20
.Apparent periodic character of the corrections 10 the eccentricity and perigee 20
Formuhc for the new inequality of longitude 24
Discussion of tircenwich observations of the moon Iroin 1S47 to 1S5S 24
Sums of residual errors 2(1
Corrections to eccentricity, longitude of perigee, and iiioon's longitude 21)
S 2— 1N\'ESTI(;.\TI()N OF P()L.\R DlSl'.VNCF, .\NI) I,.\TrrUDE.
Corrections to dei lination ilepending on errors of loiigiiude
Constant corrections lo reduce declinations to same luiHlaiiieiilal standard
Sums of errors of moon's corrected declination, given liy oliservaiions at (ireenwich and Washington.
Correction lo inclination of orliit and longilude of node
32
32
34
36
^ 3.— Al'XILI.\RV T.\UI.i:S FOR FACILITA'riNt; ITIK COMPUTATION OF TDK CORRECTIONS
HANSEN'S "TAHLES DE I,A LUNE '.
TO
Summarv of corrections to mean and true longitude of the moon from Hansen's Tables 37
E.\planation of tables for applying these corrections 37
Example of the use of the tables 411
Corrections to the Ephemeris ilerived from Uaiisen's Tables of the Moon, for Greenwich mean noon of each dav,
from 1S74. September I, to 1875, January 31 41
Tables I, H, HI. the arguments,-, /),../, A', 11 45
Tables IV, V, VI, secular and empirical terms 4f)
Table VH, lernis of luean longitude 47
I'able VIII, terms of true longitude ^ . 48
Tables IX, X, factors lor reduction lo longitude in orbit ; and for correction of latitude and reduction to ecliptic
longitude 50
Table XI, factors lor convertin,g small changes of longilude and latitude into changes of right ascension and decli-
nation 1:1
.tk%'^'w^ '.: ^ '"'I
6
INTRODUCTORY NOTE.
6
Wiicii llie proItliMii (iC iililizin",' llie oltservalioiisorocciiUiitidiis at (he sovcnil Transit,
of V<;iHis siatioiis, so as to (Ictcriniuc tin; loiigiliidcs of tliosc; .slalions willj all atlaiiial»l<!
acciiiacy, was presented to the Comniission on the Transit of Venus, it was found neces-
sai-}' to make a careful detcrniinatiou of the errors of the lunar ephenieris liefore an
entirely satisfactory solution of the prohleui couhl he atten^pted. The Secretary of (Ik;
C'oinniission was therefore charged with this work, most of the computations on which
have been made und<'r his direction l»y Mr. 1). 1?. Todd, computer for the Commission.
Wasiuxoton, M<nj 25, 1876.
I f
mu-^.
CO kR i:CTI()NS '!■() 15 !■ A I' I' I. I !• I ) TO 1 1 A N S !•: N ' S I'AIU.I'.S
Ol" 111 !•: MOON.
m
iN\Ksri(;.\ii(i\ ()!•■ EkK">Ks (»|- i,f)\(;i riDK.
One (if llic mosl iin|H»i-taiil npcijiliuiis in cniiiicclidii willi llic oltsciviilioiis ol llir
Iraiisil of Venus is fin- acciiriilc (Iclcniiiiiiilioii (»!' llic loiiniliidcs nl llic sir.lioiis. Mmiiv
ot" these slaliniis arc so liir leiiioved IVdiii lclci,napliic eoiiiiiiiiiiiciifioii llial I lie loiiiriliMles
must <lc|»eii<l iiiaiiilv on llic moon. Dclerniinalions of loii^nliiile iVoiii iihkiii enlniina-
tinns an; lonnd Ity e\|ierieiiec lo lie snhjerl lo e(Mislaiil eirurs w liieli il is dillieiill lo
(lelcrinine and allow for. Il was llienjlore a pari of llie policy ol" Ihc Aiiieri<'an Coiii-
niission lo depeii<l on occnilalioiis rallier lliaii iip(ni moon enlminalioiis litr lli"dclermi-
na1i(»n ol l(Mi<riliides. The reason Tor lliis course is, 1 lial llic disappearaiwe ol' a slar
hohind Ihe linih <d' tin' moon is a sudden phcnonicnon, llie lime ol' wliieli can alwavs he
lix<!d williin a fnielion of :i second. If llic eplicmeris ol" the moon and s!ar were cor-
recl, and ilit^ trisk oi iiic (uriiM i ,i peii'i!cl ciirie, llic lon^nlndi' could We dclciniincd
from Ihe oc(Millalion wilh Ww. saiiu; dctirce of acnirac}' llial. the pheiMniienon <'oidd he
ohserved. 'J'lie <picslion arises, how rarlhcse sources ol" error can he diminished. The
iiie(|iialitics of llu^ lunar surface liniii a source of error w liich il is iinpossilile lo avoid,
hut which is comparalivcly innocuous when many oh.servalions arc made, since Ihe
errors will he purely accideiilal, and will llicrcfor(^ he eliminalr'd from llic mean ol a
great number of ol)sorva(ioiis.
Tlio position of llu' slar ran ho dclermiiied hy meridian ohservalions wilh alinosl
any rerpiired degree of accniiu'y. We have, llieii. only lo see how liir Ihe errors of Ihe
lunar c|)hem('ri.s can he diminished: :iiid lo reduce lliese errors lo a ininimaiii is the
ohject ol the present paper.
Hansen's lahles are tak<Mi for this purpose, lieeaiisc I here is reason lo heliev(! that
the porta rhat ions on which these tahles are ioiinded are, in the main, exlreinelv
accurate; more accurate and C(niipl»(le, in faci, llian any others which liiivc heen
tabulated. Still, before they can be used lln'the purpose in <pieslion, a number oi' verv
important corrections arc rcrjuin'd, which wi^ may divide inio two classes, — {'(U'lcclioiis
to the theory, and lo the elements.
It is well known that Ilanscm increased all tln^ iterturbations of his tables by the
consliuit factor 0.0001544, on account of a supi)osed want cd' coiiicidencf! between the
m
tf'iilcr ul' lliriin; iiiul (lu! ccnttT oC ji^nivify of llic iiioon. I liiivc sliowii tliii( iriiitscii
fails to Niisliiiii tills |)usifi(iii, and that tlii'ir is no ^rtind reason to snpjtosc that tlic moon
(liirt'rs iVoni any other of tin; heavenly bodies in this respeet.* Oiir lirst course would
(liorefori! Ite (o diminish all of Hansen's ineiiualities by this liietor, \ver(> il not that, tlnro
are reasons why each of the two greatest, pertnrltations of the moon's motion, — the evec-
tionand tin; variation, — shonid Ix; found lar;;er from <d)servation than he fonnd them iVom
theory.
ErvctUm. — Tlio evection has the eccentricity as allictor; the value of the otht-r
liictor being nearly 0.4. if, then, the ad(»|»ted ccciMitricity of tin; motm be erroneous,
the computed evection will be erronc(nis by four-tenths the anninnt of the erntr. Now,
by reference to Hansen's ^^ Jhii/ri^inii:; t/ir thioirlischcn licircliiniiii; ilir in ihii Minulhi-
ffin aiigiintii(tl('ii Stiinini;c)i"\ (pa,<fe 173), it will In; seen that the eccentricity adopted
thronjfhout in the compidation of the pertnrbatiims of tin; moon is less by o.ooocxi7^
than (he vahu! he linally fonnd from observation, and adopted in the tables, iiiid lie
nsed tim latter valne, the Ihecn'elical evretion would have been i^reuler by Ibe liactinu
.000007^ ,,,, ,. , ,11 1 I . . , , I •
rro.oooi^V llie lac(or actualU used liriiiif ').(.)( K)t s 4), tlic t'\('<'li(iii, I liiis lu-
.0549C)()S
creased, is t(»o lar<(e by only (J.00002 i ol its entire amount, or o".o(). ( 'ousc(|Meutly,
the tabular coeltieient of evection should be diminished by this amount. Precisely the
same result follows, if we adopt Hansen's view of a separatiim ot' the eenti'rs of liy^ure
and irravity of lh<- moon: and llauseii himself is led to il on paiie 1/5 of the work cited,
only instead ot' o".og, he says, "kein vidles Zelmlheil <'iner Secuutle.''
Vdiitilion. — That th(! coellicicut o! varialinu residtinjj; from meridian obseivatious
will be jrreaterthan the actual coelilcienl may be anticipated from the litHowinLM-ou-
siderations. The iueipudity iu ipu-slion attains its maxima and minima in the moon's
octants In the lirst octant, wr have a nniximnm. The ehmgalion of the nu)on from
the sun is then alxMit ,^''; ami the observed position ot the union is mainly dependent on
observations of the first limb made in tlnMlaytime, wlntn the apparent semi-dianicter of
the moon will b<; diminished by the brilliancy of the surroundiiii; sky. No account of
this diminntjoii of the iij.parent semi-tliameler beiii<; taken in tin; reductions, the semi-
diameter actually applied is too lari^e, and tin; (diserved right ascension of the moon is
al.so too larg(\
When th. moon reacdies the third octant, the valne of the variation attains its inin-
imiim. The moon then transits at 9'', and the meridian observation is made (Mi the lirsl
limb, while the apparent s«Mui-diametcr is increased i»y tlu; i:radiatioii coiiseipuMit upon
the contrast betwecMi tlu; moon and (Im sky. The result will b(! that the observed right
ascension will be too small.
The same causes will mak(! the observed right asoMisioii too great in the fifth
octant, and too small in the sciventh. 'i'hese positive and negative errors of ol)served
right ascension correspond to the limes of maximum and minimum ell'ects of variation
ill iiicriMising the longitude of the; moon. Thend'ore, the observed variation will appa-
* J'rociMMliiijfH of till) Aiiiitricitri Associiitiuu for tliu AdvnucoiURDt of yuiencn, i86S, — Hilliiiiaii'H Anieriuati Journal
of Seuiiicn, Noveiiilier, iS6S.
t Altliaiiillniigvii tk>r iiiiiUii'matisvh-iiliyHiMclien CliiSHe dor Krmi);Ucli-8iivhHiNc1icii GusellHclmft der WiMoiiHulinften
Baud vl.
rciitly 1»(! Iiiru[<'i' tliiiii llic iicliial vniijitiitii. wlialcvcr lliis iniiy lie. 'I'liis secno a niiirli
more natural and [triiltaMc cause tor tlic a|)|)an<nt excess of tlw ohscrvcil over tlic theoreti-
cal perturlmlions tlian tlial assigned l»y Hansen. Hansen's factor onlv increases tin- coetli-
cient in (jin'stion l>yo".3;,; Init it seems prolialile lliat llie variation derived from obser-
vations alone woidil he yet lai^'er than Hansen's increased variation. In tiict, in iSi);, I
tlinnd, hy coni|tarin<,f the errors of the Innar epliemeris when th<! moon cidminated at
dill'erent times (d' the day, that the eireet of llie <frrater irradiation at niyht was very
8tr(Hii,dy nuirked. Dnrinii the linir years 1X62-65 tlie mean I'rrors of the iaiiles in
right ascension at diilerent tinn's of day werr as follows:*
I.
Heforo snnsd — o. 1 54
A Her briifht. daylis,'ht in the eveniiii;' — 0.093
l>el(ire liriijht ilayliij;ht in tli morniiiif- . . -fo.ogi
After sunrise -|- O' ' 5.1
In the dilli'rence hctween the resnits lor e;i' h limli, the ellect of increa.sed irradia-
tion seems to he o".o6.
The only icmaininu: term which is larire enoniili to lie materiall.v all'eeted liy the"
increasf! in (|neslion is ihe annual equation, ol' wlii(di the increaM is o".io.
A ii[lance at the errors (d' Hansen's taldes, ^fiven liy nn-ridian oliservalions, will show
that, the errors ahont the time of lirst (|narter, and, indeed, dnriuij; the first half of the
lunation, are in the nu'an h'ss l»y helween 3" and 4" than dm-injj; Ihe seeoml half.
Ilence, either the semi-dianmter, in- tlw! parallactic e(|natioii, or liolli, an^ loo larye. The
parallactic etpnition nse<l hy Hansen citrres|)onds to a value .S".9i6 for the solar paral-
lax, which value is too larir(> hy prohaldy not much less than o".io. The result
which I deduced in US67 from all tin; really valuahle data exiani was .S",S4,S ; and Ihe
determinations \vhi(h have since been made, when revised with the h.«^t data, seem to
indicate a diniinnti(Mi of this value rather than an increase. These indications are, how-
ever, a.s yet, ii little loo indelinil(( to predicate ariythinif upon. I shall I heretore con-
tinue; to n.se S".84S, which will dindnish Hansen's value hy o".o6S. The; correspond! njf
diminution in the! ]trin<'ipal parallactic term will he o".()6, while there will he two other
terms to receive a smaller dimiiiulion.
This correction will still leave a diH'erence (d" ahout ::" helween i\u\ results from
the first and second limbs, which will be accounted for by an error of 1" in theado]»ted
semi-diameter. This correction to the semi-diameler is a priori tpnte probable, as Ihe
improved meridian instruments of the present lime give a .send-diameter of the sun 1"
less than Ihe older ones from wliieli the diameters adopted in onr ej)hemerides were
derived. It is to Ik; expected that Ihe .stni' diumeler of the moon will exhibit a sim-
ilar apparent diminution.
From a note in I lansen's Ihtrleginis!; (|>age 439), it w ill be seen that one of the terms
in the true longitude has cre|d into the tables with a wrong sign. AscMnployed in lhetai)les,
and given on page 15 of the introduction, it is, -f o".335 sin (25- — 4 "•' + 2f.)— 40').
As revised in Wwlhtrlegnng, it is — o".285 sin
Theretbre the tables need the correction — o".62 sin
* Iuveiitig.-ktiou of the Distance of tlio Sun, p. 24.
2 M
'I
w
Kl
?
^4'
10
The f«»llo\vii)g is alist of llic roriTctinns wo Imvo so far deduced to Hansen's tables.
Tliev should in slri(;lness l)e applied to Ihe mean longllude, or '^Argiin/pn/ fondu/ncndir,
■l»nt they may without serious error !)(> applied to IIk; true; lonyiliide.
Put
J>, the argiunent ol' parallactie ineqnalily, or mean elongalion of the moon from
the sun ;
iT, the moim's mean anomaly :
g', the sun's mean anomaly ;
fo, the disfanee ol" the moon's perigee; from the; aseending node;
&>', tiie distanee of (he sun's perigee from the same node.
We then have
and the correetions in (piestion ar
■ 0.96 sii
0.07 sin (P
J) — n _ g' -(_ f.) .
M
4-0.96 sin 7' ^
— IT ) ■ rnvnUnetn- U'niis.
— O.I ;, sin (/>+-') >
-{-0.09 sin ii' Anniinli'iiinlH.n.
0.33 sin 2 /) Viniall.;!.
— o. 10 sin (3 D — ir) f:,',rii„ii.
— 0.63 sin { ^ 2 — 4 i,'' -|- 2 r.> — 4 m'
Ai'riilental error.
The fourth and filih terms of this expression lia\e the elfeel to remove the iiierease
wliieli Hansen applied to his inetjualities on aeeonnt ol the jtosition of the eenter of
gravity of the moim, while the sixth is the residt of the slight error of the eeeentrieity
wliieh lie emidoyeii in eompiiting the coetVici(!nt of eveetion.
In comparing with nn'riilian (d)servafions which have l»een reduced without any
correction to the apparent semi-dianiet(,'r depending on the time of day, the e(»rreelion
of variation may also ht; omitted, since a yet larger apparent correction, having the oppo-
site alircltraie siiru. will r<,'siilf, from the apparent variations of that semi-diameter, as
ulreaily explained.
As regards the possiUh; correct ions to the elements of Hansen's tables, it is t(» he
renuirked tlnit that investigator did not avail himself of the elements of tin; lunar orhit
deducefl l>y Airy from thcM Jreenwich ohservations between 1750 mikI 1S30, but <dttaiiied
liis final values of the elements by a comparison of his own. ()!' the nature and t>xtent
of the observations thus employe<l, we have no details ; but it is not likely that more
than a very small fraction of flw' entire mass of ol)servations was used, and iti can then;-
fore hardly bi' expected that the elenients were det('rmined with the last degree of
accuracy. Any error in the motion of the perii;ee 01 node will constantly increase with
the time. It', in addition to this, we rellect that Ihe meridian obs<!rvations of the lasj
twenty yejirs are lin* more accurate than llios<! Hansen had at his dispctsal, it will not
seem at all surprising to liml (piite sensible enors in Ihe present longitudes of tlu! lunar
perigee and node as derived !»y Hansen. Our lu'xt step will therefore be to d<;termine
■Wi';
11
■i
m
wliat corrections to liuiiscirs clciiiciils arc iiidiciiti'd l>y tlir rccfiit oltscrvatioiis ol tlic
iiiooii made at (Jrcciiwicli and Wasliiii^'loii .since iSOJ, a period duiinu wiiicli liolli
series of oliservations aid carefnllv conipan-d witli llansenV taldi's.
Tlie general ideas on wiiicli liie present invcslii,'atiiin ot liie>e, correeticni.s is liased
are these: tile' errors ol' liie moon's laiuilar l(Hii,nliide are ol' Iwo classes, — a progressive
correction, wliicli ap|)arently increases nniHtrmiy \\ ilii tiie lime; and errors ol short
period, tin' principal ones of which go llironLrh their |teriu,l dnriiii,' one revolution of the
moon (»r less. In determininij; the errors (if the (irst class I'roni oh.servation, those ol
IIh^ second class may he reij;arded as accidental ernns, the eli'ect of which will l>e elim-
inaled from the mean of a larire nnml>er of oliservations. Since, in a .series ol ohserva-
lioiis e.\l('ndini( (hronifh a mnnher of years, tin; maxima and mininiaof each term ol
short period will tiill indiscriminati-ly into all parts of all the other [M-riods, each periodic
c()rre(;li(ni may he determined as if the c[]\'r]s of the others were |»nrely accidental
errors. At the same time, as the elimiiialion of eaeh periodic ernn- from tin; ma.xima
and minima of all thi; (»th(M-s cannot he complete in any iinite tinn', it is desirahle that
each periodic cm-rcction of sensilde maiinitnde which we can determine beforehand shall
he applied to the rcsi<hials hcliirc the laltcrarc used to determine thi' corrections toth»!
elements.
The corrections of the elcnu-iits of loimiliidc have been made to depend principally
tii)oii the observed riulit ascensions, in>tead ol rednciny the oliserved ernns ol riiihl
ascensi(ni and polar distance to errors nf hni-iliide and lalitnde. Thi- reason lor this
course is, that the apparent errors of pcdar distance, alter correcting them approxinnilely
for errors of the elements easily (h'terininetl. will aii>e principally from ernn,* of obhcr-
vation, and not from errors of I he taldes. In tact, I lie niisei vat ions of the moon's declina-
tion are sometinn's aU'ected with accidental errors »[' ii niaunitmle which it is ditVicidt to
account f(»r, especially in the case of Washinylon. (Irantinjr that the moon moves in a
plane the position of which can be very accurately determined, we have at\erwurd only
to determine the moon's |)()sition in that plane, and this ciin In- (hnie from an ob.scrved
right ascension almo.st as well as if we had a directly observed loni,ntnde. The longi-
tude thus determined will be less likely lobe aU'ecled with systematic errors than il we
suppose! the position entirely unknown, and chaiii:e the einns of right ascension ami
declinat'on to errors of lon^^ilude and latitude, without regard to the po.ssihle constant
errors of the ineasure<l declinations.
Foriiiula- for expressing the longitude and latitude of the nuton in terms of the
lunar elements are given by Hansen in a posthumous memoir.* The following terms
are sullicient for our pres<;nt jiurpose :
Tut
/, the mooifs hnigiiude in (uiiit :
0, the hmgitude of the ascending node :
/, the inclination of the orbit to the ecliptic;
(t,^, the moon's right ascensimi and declinalioii ;
cj, the obrnpiity of the ecliplie. __^__
• II.)b«n- ili(! Diirstolluiii; <lrr Kiaili'ii AiirstciKiiiiK inid Alnv.ic lnin- drs .MoiuleH in Kmictioii der LiiiiKo in dor Itnlni
und d.T Knotcniiint;!'. Ai.!iiiii(nniii;..ii d.i KiinislicIi-S.'i.liMsili.u (J.stlls.liiifl del- WlNHt-nHcliuncn, ltd. x, No. viii.
12
■i^^^^
Wo tlieii have, apin-oximalcly,
«_/_2'^.5Hii2/- i".! sin(2/-0)+ I ''.I sill
Mill 5 zz isiii ft) sill / + cos ay sin i sin {1 — 0)
— 0.40 sill / + 0.08 sin {1—0)
Tiie (lifleiciilial co-etHcionts derived iVoin tlicse cxinvssioiis are,
(la
dl
da
do
da
— 1 _ 0.037 eos {■?- I — 0) — 0.087 cos 2 /
— 0.018 cos 61+0018 COS {2I — O)
-'*- - 0.2 1 sill — 0.2 1 sin {2I—O)
dl
cos (5 '^'^ = 0.40 cos / + 0.08 COS (/ — 0)
dl
— (^0.40 + o.oS COS 0) cos / + 0.08 sill sin /
cos«5 '^'^ =-0.081 co>i{l-0)
do
cos S "-: — 0.92 sill {I —0)
dt
From the first three forniulie, it, will bo seen, that the mean error in ri<rht ascension
is very nearly the same as the- mean error in loiiiritnde; the i.eriodic corrections lieing
siiitposed to he eliminated from this mean.
The investigation of the corrections fnmi ohservations is now made as luHows :
All the apparent errors of the tables derived from the meridian observations at Green-
vviehaiidWashingtmi since 1S62 have been collected, arranged in the order of dates
and the mean taken for each year; observations of the separate limbs being kept sepa-
rate. The mean error in right ascension for each year is as follows:
Apparent cirors of Hansen's tahles in Jl. A.
Greenwich.
Diir.
W.ishington
Mean.
Year.
I.
II.
I.
II.
Dim
I.
11.
Mean.
1862
"
"
It
"
"
"
- 3-('
- 0.6
— 2.1
l8()3
.
.
- 2.3
+ 0.5
-- 0.9
1S64
.
.
. .
— I.O
+ 1.3
+ 0.4
1865
— 0.2
+ 30
3.2
+ 0.3
+ 3-9
3.f>
0.0
-1- 3-4
+ 1.7
1 806
+ 1.2
+ 3-6
2.4
+ 0.9
+ 4-5
3.6
+ 1.0
+ 4'J
+ 2.5
1867
+ 2.4
+ 5-7
3-3
+ 2.4
+ 5.3
:.4
+ 2.4
+ 5.8
+ 4-1
1 808
<♦- 2.f)
-1- 0.0
3-4
+ 2.4
+ 6.6
4.2
+ 2.5
-1- 6.3
+ 4-4
l86()
^ 3-3
1- 5.6
2.3
+ 1
+ 7-4
4.0
+ 3.4
•I- 6.5
+ 4-9
1870
+ 3-4
+ u.(,
3.2
+ 4.6
+ 7-2
2.6
-1- 4."
+ 6.9
+ 5-4
1871
+ 5-4
+ S.2
2.8
+ 5-1
(- 7-8
2.7
+ 5.2
+ 8.0
+ 6.6
1S72
+ 6.0
+ 8.7
2.7
+ (1.2
f <)■(!
3-4
+ 6.1
+ 9-2
+ 7-6
1873
+ 6.9
+ f)-4
2.5
+ 6.9
+ 10.2
3-3
+ 6.9
-1-I0.2
4- 8.6
1874
+ 8.1 +11. 4
3-3
+ 7.t
> lO.S
3.7
+ T.(>
+ M.I
+ 9.4
'J'he last column exhibits the apparent tabular errors in mean right asccnsicii, and
13
therefore iii mean longitude, ;is tlorivfd cacli vt-ar fnun all Hit- nhscrvatiuiis. Tiic siiddfii
appaicid alleralioii oi' iicaily oiio second per aiiiiiiiii in (lie mean motion ol' the moon,
exiiihited in tiiis eohimn, si'ems t<» me oni; ol' the most (jxtraordinary of astronomical
phenomena; bnt, as I have discussed if in sevi'ral |»a|iers during the last live years, I
siiall Ju no more here tlian call attentitm lo its continuance, ami to the inipossihility of
representing it by any small mindter (d periodic terms without introducing discordances
into tlie longi(n<le during previous years.
It will he seen that there arc discordances hetwetMi the resulls of the two oi)serva-
tories, sinnetimes aiiKvuiilinL' to more than a second. In delermining the correctitMis ol"
short period, it is desiralde to reduce the systennitic erntrs exteinling through each
year to a minimum ; the <[uestion whelher such error.s arc in the theory or the ol».serva-
(ioiis being indillerent. It is also desirable that in taking the mean of the r<'sults (d'the
two (djservatories, they should be nuide comparable with each other by correcting either
of them for the .systemaiic dillercnce. 'I'he e corrections, of course, oidy admit of
approximate determination, and they have been applied ea(di year to (hat observat(»ry or
that limb of tin- mocm in which, judiring from the deviations from unilltrm proirn-ssion, it
was jiidgt'd most likely that the discordance existed. The following are (he correcdons
actually applied to the .<evend clas.><es of tabular errors:
niLX'invii
Ii.
I.
II.
s.
s.
0.06 1
+ . of)
P
1)
1- O.oO
o.oO
Waslii
ngl
on.
I.
II.
s.
• s.
< f.Oi^
(J
— 0.04
— 0.04
— 0.04
— 0.04
Ifavin;.' ap|tlied th«>se C(»rrec(ioiis throughout their s«;veral yt.-ars, the Greenwich
and Wasliiuirton ob.servations were considered s(riclly comparable; and when (he mtH)ii
was (d»served a( l»oth oli.^erviitories on the? same day, (he mean of (he correc(ed (abular
errors was (aken. Tin- meiin ou(s(ainlinir (abular error \\)v each vear now becomes as
oUows :
Vear.
,'?
Vear.
.1/
YuiU'.
(P.
Year.
-!?.
1S62
— 2.1
1S66
-1 2.2
1S69
+ >'
1872
+ 7-:,
1S63
— 0.9
1N67
+ J.^
1870
+ .v6
1^7.5
4- 8.0
1S64
+ 0.4
1868
+ 4-1
1871
4- 6.6
1S7.1
+ 9-7
1S65
+ '4
'J'hese quantiti*'.*!, with the sign <'lianged, siioidd b(! considered as c(nrec(ioiis (o the
tundameidal argument, and we have (<» de(ermine (he corresponding correction (o (h(.>
right- a«'"?ii.sions which are (o be applied (o (he individual tabular ern s. To reduce
(hem ti. ."iM'ri'ctions o(" true Innirituile, (hey are (o be multiplied by the factor
I + - '' •■^•'^ i' =^ • + o. 1 1 cos i;
14
Tin; (roiTcspoiuliiiff taclor tor correct ion of riglit ascension is, witli siitticient ui)]>rox-
iniiition,
Sazn {i -\-o.i \ CDS i,' — 0.04 cos (2 / — 0) — 0.09 cos 2 /) SX
In this I'orninlu, <5/V represents tlie correction ti> the mean loiif(itn»ie, while we may
sii|)i)ose / to represent indillerentl}' the nu-an or tlie tru(! lonyitiule ; and, during a period
ol' several months at a time, we in.-iy represent tht; lonijilnde as a t'unction of g. The
valu(! of Sa has Ih'.cm reduced to a table of doultle entry as a function of if and of tlie
time. To express th(! mean longitiule as a function of if, we have
/- i/+ TT
2 / — zz 2 g -[- 2 ?r —
I 7— 2 g -\- 2 7r
My the substitution of these values, the expression tiir Sa becomes
(5a ^; ( I -f- o. I I COS g -\- A COS 2 g -\- li sin 2 g) S\
w lie re
A ZZ — .04 COS (2 /T — 0) — .09 cos 2 /T
li ■=. .04 sill (2 /T — 0) -f- .09 sin 2 TT
The lues ot rr, 0, A, and li \\)V periods of six months are as follow :
of tl
spoi
this
Yoar.
1862.0
j 1862.5
j 1863.0
1863,5
186.). o
1864.5
1S65.0
1S65.5
I S66 . o
1S66.5
1867.0
1S67.5
18OS.0
186S.5
The coeftl(
lest! sets of
idiiijr valin; <
piipcr, it is
The correct
T
()
„
22S
274
2-lS
264
269
255 1
2S()
245 ,
3'-iy
235
33'->
226
.350
216 ]
310
206 1
31
"J7
51
1S7
71
■77 1
.)2
168 j
112
15S 1
»33
14S
+
+
.05
.0(|
.oS
.03
.02
• 05
.06
■ 05
.ot
.02
.05
.05
.04
.03
B
+
+
.0(J
.01)
.04
.oS I
■ 07 !
.04 ^
.00 i
.03 ;
.03
■ "5
."3 '
.00 t
.02 I
.05 i
I
Year.
18O1J.0
1S69.5
1870.0
1870.5
1S71.0
1S71.5
1872.0
1872.5
1S73.0
1S73.5
1874.0
1S74-5
1S75.0
JT
«
„
•
153
•39
"73
129
";4
119
214
no
234
KX)
255
90
275
81
295
71
316
61
33f>
52
SSf"
42
17
32
37
23
.07
.08
,06
.01
.06
. 10
.09
.04
•05
.12
,12
.04
B
.06
.05
,1K)
.05
.09
.08
.02
.06
.11
.11
.04
.05
, 12
ient 1 + O- 1 ' t'"*'* rJ" + ^ cos 2 g -{■ li sin 2 g is next tabulated for each
values of A and li for every 10 ' t»f g, and multiplied by the corre-
if iiX. As these tables are superseded by those yivcii at the cIos(! of
not necessary to print them,
ons ol" short period, which have been actually a|»plied, are
-I-0.96 sill 1)
— 0.13 sill {D -\- g')
+ 0.09 sill g'
— 0.62 sin (2 i' — 4 5^' + 2 6j — 4 (x)')
15
The first tliroo liave been combinod info a siiiifl(> oih; olMoiihlo aii,niiii(Mit, in wliidi
tlie argunionts sire /> antl tlio niontli; Uk! liitliM-coiic'spontlinij; to i('. Tlic Icinis dcpcnd-
ent on tiiis argument nw. so small that they may l>e regarded as eonstant during an
entire month.
In |,his sain(> talde is ineludcd a partially conjectural correction l()rtlit! variations (tf
the moon's semi-diameter. The correction to Hansen's value has l)eeii assumed a.-;
— 2".o, when the moon is in the ntnghhorhood of the sun, so that iier limb is very liiint;
and as —o".4 after the dose of evening twilight. IJetween two hours of elongation
and the dose of twilight, it is assumed to increase uniformly. The sum ol' these l()ur
corrections is given in the tbllowing table :
0) —
Ul t, O
cT S S
Q
FIRST LIMR.
Jan.
14
+ 2.4
13
+ 2.3
12
+ 2.2
II
+ 2.1
10
^■ 2.0
9
+ 1.8
8
+ 1.5
I
+ 1.5
fi
+ 1.5
5
+ 1.4
4
(- 1.2
3
+ 1.1
2
+ 0.()
1
+ o.f)
+ 0.4
Full
+ 2.5
+ 2.4
+ 2.3
+ 2.2
+ 2.1
+ 2.0
+ 1.7
+ 1.5
+ 1.4
+ 1-3
+ 1.2
+ 1.0
+ o.S
+ n.6
+ 0.4
Mar.
+ 2.5
+ 2.5
+ 2.5
+ 2.4
I- 2.4
+ 2.3
+ 2.1
+ I.S
+ 1.4
+ 1.3
+• 1.2
+ I.O
+ O.S
+ n.6
+ o..
.•\l)iil. ! May. I liiiu'.
+ 2.6 ! + 2.5
-f- 2.5 I + 2.4
+ 2.4
I 2.4
+ 2.3
+ 2.2
+ 2.0
+ 1.8
+ 1-5
:.4
2.3
+
f
I- 2.2
+ 2.1
+ 2.0
+ I-S
+ I.f.
-H 1.2 i + 1.4
+ I . I I t- I . I
+ 1.0
■1-0. (J
+ 0.8
+ 0.7
+ 0.6
+ 0.6
+ 0.4
+ 0.4
+ 2.4
+ 2.3
+ 2.3
-I- 2.2
\- 2.1
+ 2.1
+ 2.0
+ 1.8
I-- 1 . 5
+ 1-4
H- I.I
+ ".'J
+ ".7
+ 0.6
+ 0.4
Inh.
Aug.
Sept.
-I- 2 . 3 ' -I- 2 . 2 +2.1
+ 2.2 4-2.1 +2.0
(-2,24-2.1^ + 2.0
■f 2.1 I f- 2.1 + 2. CI
1- 2.0 ' -f- 2.0 I- 2.0
+ 2.0 4-1.1) 4 I.S
-I- i.() 4- I.S 4-1.7
4- 1.7 I t- 1.6 I 4- 1.4
4 I. ;
4- I.I
4- I . '
4 11.9 ; 4- 1.0
+ 1-5
+ 1-3
t- 1 .0
I
-ho.S
4- o.S
-f 0.6
4- 0.6
-f C.4
4- 0.4
4- 1.2
+- 1.2
4- I.I
f 1.0
-f 0.8
-I- 0.6
4- 0.4
Oct.
Nov.
Der.
+ 2.1
4- 2.2
-!■ 2.3
4- 2.0
4- 2.1
•t- 2.2
+ 1.0
■(- 2.0
4 2.0
1 1.8
-1- I..)
4 2.0 1
i 1.7
1- 1 . 7
4- I.S
4- 1.6
4-1.5
4-1.6
4- 1.4
+ 1.4
+ 1.5
+ 1-3
4-1.4
+ 1.5
+ 1.3
4- 1.4
4- 1.4
+ 1.2
4- 1.3
-1- 1.4
4-1.1
4-1.2
4- 1.2 1
4 1.0
f l.I
4 1.1
4-0. S
+ o.<)
4- 0.9
4- 0.6
4-0.7
4 0.6
40.4
-ho.4
4-0.4
"^ r! O
lyi 4j O
S'S s
c c
IT O O
Q
4
4-
-I-
+
4-
+
4-
+
4-
+
4-
+
o
I
2
3
4
5
6
7
S
9
10
II
1 2
13
14
Jan. I Feb.
-0.4
-0.4
-0.6
-0.6
-O.S
-0.7
-I.I
-0.9
— 1.2
— I.I
- l;4
-1.2
- 1.4
— 1-3
- 1.5
- 1.3
- 1.4
- 1.5
1.7 ; - 1.7
I
— 1.9
- 2.0
— 2.1
— 2,2
-2.3
— 1.9
— 1.9
— 2.0
— 2.1
— 2.2
Mar.
April.
-0.4
-0.4
-0.6
— (1.6
-o.S
-o.S
— l.O
— 1 .0
— 1.2
— I.I
— 1.2
— 1.2
- 1.3
- 1.4
- 1.6
- 1.6
- I.S
- I.S
-2.0
-1.9
— 2.0
— 2.0
— 2. 1
— 2.0
— 2.1
— 2.0
— 2.1
— 2.0
— 2.t
— 2.1
Mav.
- ".4
- 0.6
- (I.S
- 0.9
- I.I
- 1.4
- 1.5
- 1.7
- 1.9
- 1-9
- 2.0
- 2.0
- 2.1
- 2. I
- 2.2
s
FCOND LIMB
Dec.
June.
July.
Aug,
Sept.
Oct.
Nov.
-0.4
-0. t
-0.4
-0.4
-0.4
-0.4
-0.4
-0.6
- 0.6
-0.6
-0.6
-0.6
-0.6
-0.6
-o.S
-0.7
-0.7
-o.S
— o.S
-o.S
-0.8
-0.9
-0.9
-0.9
- 1.0
— 1 .0
— 1.0
— I.I
— 1.0
— 1.0
— T.I
— I.I
— 1.2
— 1.2
— 1.2
- 1.4
- 1.4
— 1.2
- 1.3
- 1.3
- 1-4
- 1.4
- 1-5
- 1.5
-1.5
- 1.4
- 1-4
- 1.4
- 1.5
- 1-7
- I.S
- 1-7
- 1.6
- 1.5
- 1.5
- 1-5
- 1-9
— 2.0
- 1.9
- 1.9
- 1.7
- 1.5
-1.5,
— 2.0
- 2.1
— 2.1
— 2. 1
- 1.8
- I.S
- 1.7
— 2. 1
-2.1
— '* 2
- 2.3
— 2.1
— 2.0
- 1-9
— 2.1
— 2.2
-2.3
-2.4
22
- 2.2
- 2.0
— 2.2
-2.3
-2.4
- 2.4
- 2.4
-2.3
— 2.1
-2.2
- 2.3
-2.4
- 2.5
-2.5
- 2.4
-2.3
- 2.3
-2.4
-2.5
-2.6
-2-5
-2.5
-2.4
- 14
- '3
- 12
- II
- 10
- 9
- 8
- 7
- 6
- S
- 4
- 3
- 2
- I
o
c c
« o
(/I 01 o
J?E 6
a
o
I
2
3
4
5
6
7
8
9
10
II
12
13
14
-f-
4-
+
I-
+
4-
4-
, -t-
-I-
4-
■I-
4-
+
4-
■ •*'=Wrtfe«*rtSS«'HilBBWl»Slj*.i« 0tmy**'-j.
>^i.i**wm^3«w**r'
i. I
I ll
ij' 111
«tl
16
]\y tlic ii|)|)li(;i)tit)ii of tlio foroiroinir corrections to tlic errors of llic moon's taluilar
rii^lit asf-ensioii, these errors iiiiiy he supposed to he rediictMl to very small (piaiitities,
dependiiiij; on the errors of the lunar (!l(!ments, with which th(;y are eonnecleil hy the
e(piation
„ ■ (lot ^, , (In »,j , (la ^.
(U ^ (1.0 ^ (li '
the clillercntial coelticieiits having tlie vahuvs ijiven on |)aire 12. When we snl)stitnte
theso values, tiie exprcission for Sa will contain the terms
(+ .01 8 (5(9 — ,03 7 '5a) cos (2 / — <9)
— .087 6a cos 2 /
+ .018 (5 (9 cos (9
-|- 0.21 <5/ sin 6
— 0.2 1 (3i sill (2 / — 0)
If we represent the sum of these; terms l>y P, we shall have
SI =<'ia — 'D
In the investigation of the corrections to the; moon's eccentricity and longitude of
perigee, the terms of P may be entirely neglected. This arises from the circumstances
that tlu! appreciable terms of /or a arising tV'im tlie errors of these elements liavt; the
same period with f, tiw; mean anomaly, while /'contains no apprecialde pi'riodic terni
depifuding on g. The outstanding pmtion of ('ia prol)aldy averages not more Ihan »me
second or two at the utmost, so that the term .037 ('>a is (piite insignificant. The term
.018 SO may have a constant value of o".25, more or le.ss;* l)ut tlu; short period of the
term 2/ — (9, ami its inc(»mmensnral»ility witli the period of i,', permit of this error
i)eing regarded as liirtiiitoiis. The sanii' rentark ap[»lies to tiiiMcirms .0S7 S(t cos 2/
and 0.21 r5/sin(2/ — 0). 'i'he only remaining terms liave tiie jteriod of 0, which is
niorf- than (Mghleen ye:irs. 'J'lie ellect ot' these possildt; errors is tlieiell)r<' eliminated
in th(> mean correction for each year, which has been alrea<ly applii-d to the errors.
To determine the correction to tin! (•(•(•entricity and longitude of the perigee result'-
ing from each year's observations, the residuals in riglit ascension, afl(!r the application
of the three corrections already described, have becjn arranged according to the values
of the mean anomaly to which they corresi»ond. The results are shown in the follow-
ing talde, which gives for certain limits of mean anomaly in the first ctdumn, firstly, the
sum of the residuals (tal>idar iiiliiii^ o!)served) in riglil asctiusion, corresponding to all
the values of mean anomaly between those limits; and, secondly, the number of the
residuals. In taking these sums, the observations at the two ob-servatories are counted
st^paratrly, .so tl'.at when observations wen; made at \m\\\ obs(M'vafories on the same
date, th(! sum of the residuals is tak(Mi, ami the observations count 2 in the column N.
"It IH iiftcrwiinl tuiinil that the viilno of this |iii>. ■■t is only o .oS.
17
Sums of errors of tnoo)i\s corrected rifjht ascension, given hy ohserrations at Grccnmch und
Washington.
1362.
1863.
1864.
1865.
Limits of mean;
anomaly.
i
1
N.
f
1
£.!a
N.
S,!n
N.
D.ia
N.
. i
o i
o lo lo
+ 3-9
4
H
+ 21.5
10
+ I9.f>
9
+ 1-4
7
10 lo 20
+ 3.f'
6
+ 12.3 :
12 !
+ 6.1
7 :
+ 3-4
4 *
20 to 30
— 0.2
5
+ 14.2
8
+ 5-8
5 j
— <i?
10
30 to 40
+ 9-3
8
+ 23.7
II
+ 4-5
7
- 0.5
5 1
40 to 50 1
+ 2-7
8
+ 9'0
8
+ 2.6
3
- 3.f'
6
50 to 60
+ 0.3
8
+ 9-8
9
- 1.6
10 1
— I.I
6 1
60 to 70 '■■
+ S.9
10
- 4-3
7
+ '-.7
5
- 61
7
70 to So
- 3-7
4
+ 7-0
10
- 7.0
f' i
- f>.7
6
80 to 90
+ 6.7
7
- 6.7
6
— II. 2
9
- 6.1
6
qo to too
+ 3-9
6
- 3-3
9
- 3-4
6
- 8-5
7
too to ito
+ 3-9
11
- 0.4
5
- 2.1
5
— 0.7
5
110 to 120
- 6.4
9
- 3.9
8
- 30
3
- 7-5
8
120 to 130
- 3-2
8
- 3.9
7
+ 0.1
5
- 5-5
6
130 to 140
- 7-8
6
- 8.8
8
— 12.2
7 i
+ 5-0
5
140 to 150
- 0.9
5
- 15-9
8
+ 0.9
3 1
i
+ ...
5
150 to lf)0
— 0.1
5
- 18.2
9
- f>.7
7
+ 1.5
4
160 to 170
- 8.8
4
- 19.7
6
+ 2.5
6
+ 4-3
5
170 to iSo
- 5.7
4
- 9-9
7
- 5-3
5
+ 6.4
6
iSo to 190
- 17-4
9
- 33.1
14
- 8.6
7
+ 8.9
6
190 to 200
- 15-5
7
- 4-3
4
- 0.6
4
+ 15-2
8
200 to 210
- 3.S
10
— 1 .0
6
- 6.4
9
+ 7-8
8
210 to 220
— 0.2
2
- 1.9
9
- 2.9
8
+ 13 I
7
220 to 230
— 28.9
9
- 7.5
10
+ 3.<i
/
+ 5.1
5
230 to 240
- 7-3
7
- 1-9
7
+ 0.8
7
+ 10.3
5
240 to 250
+ I3-0
8
+ 0.4
9
+ 1.6
7
4 7.3
8
250 to 260
— 2.0
4
+ 7-6
8
+ II.5
8
+ 7-3
7
260 to 270
+ 1.6
9
+ 1.4
! 5
+ 11-7
7
+ 16.2
12
270 to 23o
+ 3-7
5
+ II. 3
9
' + 25-3
II
+ 7.('
II
2S0 to 2</3
+ 4-7
7
1- o.S
5
+ 18.2
8
+ 9-f'
8 :
290 to 300
- 1.3
1 I
+ 15-9
7
; + 6.6
4
+ 5.8
II 1
300 to 310
+ 3-0
i 3
+ 23.5
9
+ 7.8
6
+ 10. 1
7 i
310 to 320
+ 2.3
1 2
+ 22.6
; 6
+ 6.4
5
+ 16.4
10
320 to 330
- 2.8
5
+ 18.2
9
+ 11.6
7
\ + 14.5
7
330 to 340
+ 9-5
6
+ 1.2
7
+ 18.5
10
+ 16.7
II
340 to 3;o
+ II. 8
8
+ 7-2
7
+ 4-2
7
+ 7-6
1
7
350 to 360
+ i3.f'
5
+ 14-4
8
+ 16.5
6
! + 5-3
9
j +106.4
225
+ 222.0
2S7
1 +1S7.1
i 236
; +205.9
255
i —116.0
1
1
1
-144-7
-
1 - 71.0
j
- 46.6
-
j - 9-6
+ 78.3
+ 116. 1
+ 1593
Sm
»^»*>*~-"'*«»e!B«»«wss*<i«nBi4ii«»*'»»
18
Slims of erron of 7noons corrected right aKcension, S^v. — Contimiod.
1866.
1867.
1868.
i86g.
Limitsof mean
anomaly.
£(ln
N,
Ida
N.
XAa
N.
2, In
N.
to 10
- 1.7
6 i
+ 7.4
5 !
n
- 4.2
4
- 10.7
4
lo to 2o
- 2.5
4
- 5.0
2 1
i
+ 3-9
7
- 4.2
4
20 to 3P
- 7.5
3
- 1.7
4
- 2.5
3
- 0.8
6
30 to 40
- 7.1
5
- 7.5
3 ,
- 9-4
6
+ 4.2
5
40 to 50
- 14-5
7
+ 5.5
'
- g.o
5
+ II.
6
50 to 60
- 0.7
I
— 2.0
4
- 0.7
7
+ 5-5
3
60 to 70
+ 1.3
5
- 8.5
4
+ 2.2
7
+ 3.1
5
70 to 80
+ 5-3
6
- 4.8
3
+ 4.1
8
+ 7.7
7
80 to go
+ 1.6
6
- 3.6
I
+ 12.2
7
+ 8.0
8
90 to 100
+ 3.9
4
+ 2.6
5
- 0.3
4
+ 16.8
8
100 to no
+ 4-4
9
- 0.6
5
+ 14.9
7
+ 5-1
9
no to 120
+ 4.«
8
+ 3.9
5
+ 9.8
6
+ 8.3
6
120 to 130
- 5-4
8
-(- 1.6
7
+ 4.1
5
+ 14.5
7
130 to 140
+ 3-4
6
+ 4.1
6
+ 10.2
8
+ 7.5
8
140 to 150
+ lo.l
9
+ 1.9
7
+ 5.2
7
+ 3.1
6
150 to I Go
— 4.'
6
- 2.6
7
+ 2.1
9
+ 20.3
7
t6o to 170
+ 3-3
7
+ 6.8
5
+ ..3
8
+ 3.7
3
170 to 180
— 0.1
7
- 5-0
8
+ 0.8
7
+ 12.2
7
180 to 190
+ o.S
6
- 0.3
2
+ 12.3
8
+ 7.0
5
190 to 200
+ 5-9
6
+ 2.0
4
+ 17.9
6
+ 6.3
4
200 to 210
- 3-2
6
+ 2.8
6
+ 5.2
5
+ 10. 1
5
210 to 220
+ 0.3
6
- 1-7
4
: + 13.0
8
+ 12.2
5
220 to 230
- 5.4
4
+ 12. 9
9
1 + 4.8
4
+ 12.3
7
230 to 240
+ 4.J
8
+ 8.2
6
+ 15-2
9
- 1.3
3
240 to 250
- 1.8
7
+ 25.4
9
i 4- 7.4
8
- 6.4
6
250 to 260
+ 9-4
' 7
+ 0.9
3
'. + 14.2
8
- 3.6
2
260 to 270
+ 2.7
7
+ It. 7
6
: - 5.0
2
- >7.3
7
270 to 280
+ 9-:
4
; + 3-3
4
+ I.O
7 •
- 18.8
5
280 lo 290
+ II. 6
12
1
+ 7.0
7
- 9.1
5
- 21.4
6
290 to 300
+ 4.0
! 4
+ 0.7
3
- 3.2
8
— 13.6
3
300 to 310
+ 6 7
4
+ 16.5
7
- 8.0
i 2
- 4.8
2
310 to 320
+ 3-4
2
i + 2.3
5
- 13.8
8
- 0.8
I
320 to 330
+ 7-7
5
1 + 0.2'
5
— 10.6
9
- 4-2
2
330 to 340
+ 9-1
5
1 + 3.5
6
- 11.7
6
- 18.5
6
340 to 350
+ 10.8
6
\ - 5.4
7
. - 9-8
I
5
— 10.6
4
350 to 360
+ 92
7
- 7.2
4
1 - 18.3
6
— 2.2
5
+ 132.9
213
+ 131. 2
182
+ 161. 8
229
+ 178.9
187
- 54.0
1 - 55-9
— 115.6
1
1
-139.2
+ 78.9
1
1 + 75.3
+ 46.2
+ 39-7
19
Sum/) of crwrs of moon\s corrected right ascension, S^r. — ('(mcliidcil.
Limits of mean
anomaly.
1871
187
.
1872.
1873.
1874.
7,,\a
N.
Sila
1
2,1(1
1
N,
V
rta
N,
S,l<.
1
N.
OlO 10
- 7.2
5
- 3.2
5 + 6.5
6
4.3
6
+ 4.6
1
1
1 '*
10 to 20
— 2.2
5
+ 1.7
n 1 + 8.5
10
+
5.2
4
+ 5.9
S
20 to 30
+ 5.1
6
- 0.3
7
+ 5.5
8
+
5.2
8
+ 12.5
i f'
30 to 40
-f 10.7
8
+ 6.4
7
+ 1 1. 8
7
+
3.4
3
+ 5.1
i 5
40 to 50
+ II. 3
8
+ 16.7
9
+ 6.0
4
+
6.6
4
+ 4.4
i
j 5
50 to 60
- 7.1
5
+ 9.7
6 + 13,2
6
+
4.1
7
+ 2.1
5
60 to 70
+ I.O
9
+ 18.9
8 + 10.4
3
+
13.4
6
+ 10. 1
4
70 to 80
- 2.6
5
4- 10.2
7 1 + 12.4
8
+
'3.5
3
+ 6.6
6
80 to 90
+ 12.0
12
+ n.7
1 5 * + II. 3
4
+
15.8
7
+ 6.0
3
90 to 100
+ 10. 1
8
+ 12.5
i 3
+ 9-8
4
+
5.1
2
+ 5.9
7
100 to no
+ 10.8
4
+ 19.7
8
+ 13.0
6
+
1.5
3
+ 10.9
6
no to 120
+ 5-8
6
+ 8.2
4
+ 18.7
6
+
5.3
2
+ 4.6
4
120 to 130
+ 10. 1
7
+ 9.7
5
+ 18.3
7
+
6.1
5
- 4.7
6
130 to 140
H- 10. 1
5
+ «5.4
5
+ 0.2
2
+
3.3
3
+ 1.8
1
140 to 150
+ 18.2
8
+ 2.1
3
+ 2.9
3
+
8.4
5
- 0.8
7
150 to 160
160 to 170
170 to 180
180 to 190
+ 4.4
+■ 8.8
+. 6.9
+ I 8
3
5
3
I
+ 3.0
+ 8.7
+ 6.2
+ 3.9
7
4
6
+ 2.1
+ 6.6
— 1.2
+ 1.9
8
5
3
4
-
3.9
5-4
1.7
2.2
3
4
3
4
+ 1.3
— 10. 1
— 1.0
+ 5.0
5
9
6
6
190 to 200
+ 7.5
4
+ 3.5
3
— 1.2
5
-
6.6
6
— 1.0
2
200 to 210
210 to 220
220 to 230
+ 2.1
- 2.5
5
3
3
+ 1.0
- 2.6
- 9.3
3 - 2.2
2 +1.2
7-7.2
6
3
5 1
0.9
6.6
o.i
2
3
4
+ 3.7
- 5.0
— 16.0
6
5
7
230 to 240
240 to 250
250 to 2f)0
260 to 270
270 to 280
280 to 290
- 0.4
- 9-7
- 12. I
- 2.3
1
-.2.9
- 5.6;
5
5
6
2
8
3
- 3. a
- 9.1
- 5.2
- 4.6
- 7.1
-- 2.7
6 - 4.8
8 ; - 6.5 ^
5 - 9-1 i
5 i - 13.8
7 ; - 8.4 \
6 t — 16.7
5 '
3 :
4
8
5
9
—
3.5
7.5
7.1 1
8.6
4.3 1
10.8 i
I !
5 '
4
1
3 i
4 1
6
- «3.5
- 15.1
- 23.0
- 22.6
- I?. 6
- 9.1
4
8
5
4
4
3
290 to 300
- 5.5
4
+ 4.0
4 ! - 10.3
S
—
9.8
4
- 13.4
S
300 to 310
- 4.0
4
- 9.5
6-9.5
5
—
1.8 1
I
— O.I
9
310 to 320
- 8.7
3
- 6.6
5 i - 5-6
4
—
3.2 (
4 1
- ;.3
6
320 to 330
- 13.5
6
- 4-9
7 i - 8.5
5
—
11-3
7
- 1.4
7
330 to 340
- ..7|
4
- 2.8
7 1 - 8.5 :
5
—
0.3
3
- 4.3
3
340 to 350
350 to 360
- 3.6 i
- 8.7 !
3
5
- 1.7
+ 6.3
4
4
- 5.1 '
+ 0.1
5
6
—
9.2 '■
4.0
6 I
5 1
+ 2.2
+ 2.5
II
6
+ 136.6 ,
185
+ 179.5
203
+ 160.4 i
195
+
56.9
155 .
+ 95.2
200
— 120.6
- 72.8
-118. 6 j
-113. 1
— 171.0
+ 16.0
+ 106.7
i
i + 41.8
-
16.2 '■
1
- 75. s
Neglecting all tcrinr, multiplied hy the eccentricity in tlu; coefficients, each
ual gives an equation of the form
Jl-\- 2 sin gJe— 2 cos'g e Jtt — r
rcsKi
'"•^^BBimMn
■ r
1'
1!i
■'!
II-
or, piittirii,
the ncjuatioii will \hi
20
h zz. 2 JSe :r — 2 ih'
k z=. — 2 J(i y')Tr ■=: 2c Stt
Jl -\- h ^\x\ g -\- k ros if ■=. )\
Je nnd Jit I)i;ing tlio errom of tlio taimlar eccentricity and longitude of the perigee,
while 8e and fin n^proseiit IIk; corresponding corwct'ums.
The erpiations are now solved as if all the residuals within each pair of 20° limits
corresponded to the mean ol the limit, — that is, as if all hetween 0° and 20° corre-
sponded to i' zi 10° ; those l)etween g zz. 20° and if =. 40° to g — ,30°
and so on.
If.
then, w(! put
gi = 10'
Vi, the SI
•r-i
zz T,o^, etc.
)f all til
dual;
IS III any one year corresponding to g =zgi;
Hi, the corresponding iuinil)cr of observations;
A-.zz sin gi]
Ci zz cos gi :
the normal equations for det(!rmiiiiiig SI, h, and k, by least sciuarcs, will be :
i^'n,) Jl+(^„,s,) /t + (^'«;6-..) kzz^u
(^' ,U .V.) Jl + {:^ lU sr) h + {2 n, .s, 6v) k = 2 Si t-i
{2 „i d) Jl + {^ «,. .sv d) h + (^' », en k = 2 d Vi
The formation and solution of tli(;se erpiations for each year give; IIk; following
values of the oiilstiUKliiig errors of llie lunar eleiiienls l()r each year:
1862, /<zz + o.o4
1S63, —0.64
1864, — 1.07
1865, — 1.03
1866, —0.47
1867, —0.93
1868, +0.34
1869, 4- 1.67
1870, +148
1871, +1.65
1872, +2.15
1873, +1.91
1874, + 1.92
The periodic character of these residuals is very reiiiarivable, indicating, as it does,
either a hitherto unknown inerpiality of tlie moon's mean longitude, having nearly the
same period with the orbital revolution; or one of the eccentricity and longitude of
perigee, having a period of between tifteen and twenty years. To investigate this in-
e(piality, we shall assume that each value of h is of the tbrm
kzz-Y 1.23
+ 1.78
+ I -09
-0.15
+ 0.10
— 0.36
— 1.46
-1.56
•- 1. 14
— 0.36
• — o. 1 2
+ 0.16
+ 0.60
and each value of k of the form
h — a sill (yu -{- ut)
/?; + a' cos (yu' +»'/;),
21
3S,
he
of
in-
h k a it' fi, fi\ », aiul n' Iumms,' m.km.wii (unuilili.-s (.. I..- .Ict.-iiMin.'.l, an.l / tin- tun.,
in v,>ars fmni unv assumrd ..,m.H.. Wr slmll tak.. for tl..- qM'«''' tl'"' "n.l.H.' -'I t "■
pori.ullhn.UKli wliicl.tl..-..l.s.Tvati..ns.'xto...l;tliat is. 1868.5. If, tli.-ii, w rrpivscnt, tlw
thirtiMMi values of // and /.• in cliroiioloi^ical order by //-,„ /'-.^. ^'u' /'-r..^'-... ■ ■ • • .
k„ the e.umtioMs .>f (■on.litioi. for h and k resix-etively may Ix^ put into tlw lorn.
h; zzh — a sill // cos in — n cos /< sin / n
A-,. = /•+«' cos n cos / n — n sin /( sin i 11.
Rcuanlinii A, /.-. « sii. //, a cos /^ «' sin n, an.l «' cos /< as tin- unknown .,nantifi.-s,
tlic normal ecinations for determining thes.; (piantiti.'s ar.' :
(i) From the vului's of li,.
13// — (2' cos in) a sin /< — ^^i
- (V OS / «) h + (^' cos- /• n) a sin /« =: - :i' A, cos i n
(>;" sin-' i 11) a os /< — — ^ I'i sin ' "
(2) From the viiiucs of A',.
13/,- + {2 cos i w) «' cos /<' =: ^ ^j
(^' cos i «) ^- + (^ '''"*' * ") «' ^■"'^ ^'' = - ^'^ ****** * "
.^' (sin'-' i 11) a' sin /<' z= - :^" A'.- sin / «
It will 1..^ ol)s(M-vod tlnit all the coefficients having as a factor c_itl.er ^' sin I n or
2 sin i n cos i n vanish. , , 1 ^
The value of a apparently is not r.-a.lily determined dire.-tly l.y least .scinares : wo
shall therefore assunu. sevral values of this .p.antity. an.l as..-rtain l.y which vahn.- tlu,
n.u.litions can l.est i,.. satislied. Tin- lollowing are th.' ahhreviate.l values ol the pur.'ly
triiionometric sunnnati.nis :
sin 6i n
2 cos f II =r . — V — = c
8111 2 n
1% sin n + sin 13 « _
2 cos- in— - - 7 — -^ 1
2 sill n
^-, . „ . 13 sin » — sin 13/; _
2 sm* in — - . — *i
2 sin n
It we solve the preceding equations, and put, for brevity,
c
C -
C\ =
1 3 Ci — c-
13C1 — c-
c 13
13 c, — C''
the resulting expressions for the unknown cpiantitics are:
// = L\2hi— C^'//.. cos in
« sin /( = C2hi—C^ hi cos in
a cos yu zz 2 hi sin i n
k — C\2 ki — C2 ki cos ( n
a' cos /i' zz — C2 ki + C^ ki cos i n
a! sin fj! — — - '^ h si" ^ «
«i
»>»>
'I'lii- pciidil (»r/Miml It lies prolml)!}' Ixifwn'ii liriccii iiixl twenty yearn, whi(^li would
miikf the value nl" //, or tli(;iuiniml motion ol' llie iiiet|imlity. lie hetweeii iS'^ and 24".
Tlie lollowiiii; iire tlie values ol' the various t|iiaiitities *le|>eii*liiig on n tor the tlitVerent
values of n between these liniitri :
logf
logfi
log S\
logf
log t'
log C,
0.756
0.715
0.S93
9.213
9.17a
9.571
0.705
0.707
0.898
9.097
9.099
9.506
0.644
0.705
0.900
8.977
9.038
9447
0.577
0.709
0.897
8.858
8.9()0
9. 395
o.4<)8
0.718
0.891
8.734
8.954
9.350
406
0.731
0.88a
8.604
8.939
9.31a
0.391
0.747
0.870
8.453
8.()0<,
9.276
0.143
0.765
o.Ssf)
8.275
8.897
9.346
n
S4|iin>M
£ hi cos 1 n
S/t<slni»
iki cos in
a
18
+ 11.48
+ I . '/>
- 4.66
- 4.66
'9
+ 11.66
+ 1.52
- 4. 08
- 5.04
ao
+ ir.78
4- 1 .09
- 4 69
- 5. -to
ai
+ 11.83
+ 63
— 4. 08
- 5.73
23
+ 11.81
+ 0.29
- 4.00
— . 04
33
+ 11.73
— o.oS
- 4.02
- 0.33
34
+ 11.58
- 0.44
- t.57
- 60
25
1- It. 37
- 0.78
- 4-5"
- 6.80
The precediiisf e(|natioiis now jrive the following separate values of the unknown
quantities, eorresponding to tlu; various assumed values of «:
n
h
a
f
*
. a'
f'
1,
,
„
.
18
0.73
J.53
164.0
0.73
l.Sl
160.8
19
0.69
1.53
165.2
0.61
1. 71
159.7
20
0.66
1.53
166.3
0.49
1.62
158.5
31
0.63
1.54
167.3
0.39
'.53
• 57.2
22
0.61
>.55
168.1
0.31
1.47
156.0
23
0.60
1.57
i6q.o
0.23
1.42
154.8
24
0.58
'.59
169.8
0.17
1.39
'53.6
25
0.56
1. 61
170.4
O.II
1.36
152.6
There can he litth; serious doubt that in the case of the pnisent inerpiality the
theoretical values of /j. and /<' should he the .same; and it is also |)rol)al)le that those of
a and a may Ix; suhstantially identical. Th«! small ditrerenccs between the values of a;
and a' and of // and //' add so much weight to this probability that we shall make
2:^
icr sdliition of Hie (■(|iiiilions on tlic siiiti»(isili(in llmt a' — ir athl /<' — /i. 'I'
niiotl
null i'((iiuti()iiM tliun Ix'coiik;
If nnr-
13 // — frtrsin ft = ^/i^
— ck-\- iT,am\ /.i z= — ^ hi
cos / n
-:>■/,■.
1 3 A -f- ca cos ju z= ^ /{i
c A' 4- 1 3 a cos // — 2 ki
The solution ot these equations is:
cos I
n-:^h:
<ni / n — N,
sin / )i
•3 — ' 13' — '"
//.
'3 — ' '3 — '
a cos//— ., '^ .^«S., — ,/ ., :i'/-i
A conipiiris.Mi of tiu; separate solutions ol the e(|iuilions in h and /- shows that tlu!
valui .tf n which best satis(i(!s the (Mtuditions lies hetweeu 22'' auil 25^. The values
of /(, /•, a, aiul /i were therotlu-e (hiriviid only from the last etjuations for tin: last four
values of ti. For each of th(!S(! separate valm-s of «, tin; eorrespondinir valin-s of //, au<l
ki \v(!r(! C( uiputed from the formula'
//, = /i — (r sin (/< -f- i h)
lii zzk -\- a cos (/< + ' «)
ii) which, it will be rcniemhcred, the index /' is simpl)- the nuniher of the year I'rom
1868 ; so that we have,
For 1862, iz=i — 6
For 1863, i zz— 5
etc., etc.
These computed values of //; and /<■, were then compared with (he valiuis derived
directly from observations, and given on page 20, and the sum of the s(puires of the out-
standing residuals was taken. Tlu; valutas of the unknown (piantities, together with
the sum of the squares of the residuals, are as t()llovv :
22
23
25
+ 0.06 I + 0.34 ' 1.54
+ 0.63 1 + 0.27 1.52
+ 0.61 ! 4- 0.20 I. 51
+ 0.58 -t- 0.14 I 1.49
/'
V
lf)1.2
3.207
161.3
3- '70
I6I.5
3.246
l6i.7
.1.441
The sum of the s({uares becomes a minimum for n zr 22^.8, showing a j)eriod ot
the inequality of 15^.8, witli a possible error of a year or more. The formula' for //< and
ki thus become :
A. = + o".64- i".52siu(i6i°.2 + 22^.8/)
A< = + 0".28+ l".52COs(l6l°.2 + 22°.8/)
9A
S-*-!
^^
from whifli we liavo flio following comparison of tli(! compiilwl iiiid ol)servc(l values ol
hi aiitl l\:
Year.
h
/•,
C.
0.
O.-C.
c.
0.
0.
-C.
1862
+
O.OI
+
0.04
„
+ 0.03
+
1.67
+
1.23
_
0.44
1863
■ -
0.48
-
0.6}
0.16
+
1.32
+
t.:s
+
0.46
1864
-
0.79
1.07
— 0.28
+
o.So
-1-
1 .09
+
0.29
1865
-
0.8S
-
1.03
— 0.15
+
0.22
-
0.15
-
0-37
1866
-
0.74
-
0.47
+ 0.27
-
0.38
+
O.IO
+
0.48
1867
-
0.37
-
0.93
— 0.56
-
0.85
-
0.36
+
0.49
1868
+
0.14
+
0.34
+ 0.20
-
I. If)
-
1.46
-
0.30
1869
+
0.74
+
1.67
+ 0.93
-
1.23
-
r.5f.
-
0.33
1870
+
1-33
+
1.4S
4- 0.15
-
1.07
1. 14
-
0.07
1871
+
1.80
+
1.65
- 0..5
-
0.70
-
0.36
+
0-34 i
1S72
+
2.09
+
2.15
+ 0.06
-
o.iS
0. 12
+
0.06
1873
+
2.15
+
1. 91
- 0.24
+
0.42
^-
0. 16
-
0.26
1874
+
1.98
+
1.92
— 0.06
+
1. 00
+
0.60
0.40
Tiio ]frol»al)le residual for each year is d' .2'].
We have supposed tlie hypothetical inequality of longitude to Ik; of the form
idv zz lii sin g + ^'i f^**'"* - ■
Sub.stituting in this the periodic part of //(and /,, and replacing / by /, which now reprc-
sent.s the time in years from 186S.5, it becomes:
^vzz i".52 sin (g'+25i°.2 4- 22^.8/)
or
Jv= i".52 sin [5-+ 22°.H ( r- IS57-5)]
The entirely unexpected character of tlu; periodic term thus brought to light ren-
ders its verification by a longer series of observations very desirable. For tliis purpose,
we need comparisons (tf ol)servati()i.s previous to 1862 with Hansen's tables, because
none of the older tables with which comparisons have been made are accurate enough
for the purpose. Now, the Greenwicii Observations for 1859 contain, as an appendi.x, a
Cf»mpari.<on (»i tlu; longitudes and latitudes from Hansen's tal)les with Greenwicii ob.serva-
tions from 1847 to 1858 inclusive; ; and I have; utilizeil the comparison of the longitudes
derived frciiii meridian observations in the Ibllowing way :
A list of limiting dates t tenths of a day was made out, including the whole twelve
years, and sliowing lietween what dates the moon's m(;an anomaly was found in each
sextant. Tiie sum of the errors in longitude given liy the meridian o'oservations was
then taken during the period that the anomaly was found in each sextant. None of the
corrections found in the first part of this discussion were aj olied, for the reason that
mo.st of them could l)e treated as accidcnital errors, and the means could be taken so as
nearly to eliminate the effects of tlie larger ones. A specimen of the form chosen is
here given. Under (;ach of tlie several values of g, given at the tops of the several
,
25
re-
oulmiiiis, is .shown, firstly, liif dale al wliic.li a liail thai particiihii- vahic: and, scroiidly,
the sum of the residuals in h»niritu(h' dnriuir tlic period of 4''.6 Itctwct'ii that dalo and
the one iiiixt lidlowing, logetli(3r witli the; inind)cr of tlic i-jsiduals, tlic hitter iieinix in
small sidiserijit tiirures.
I. in.
V<1,.
M:ir.
.\|.ul
May
(line
[illy
Auk.
' Auk.
j Sepl.
; Oct.
I Nov.
Dec.
Jan.
Fcl..
Mar.
April
May
May
I u ru-
in ly
Auk.
Sc-|.t.
f)<-t.
Ndv.
Dec.
= o +
I").li- 2.1)1
Id. I - 1 .(),
>5.7 . . ;
'2-3 . . !
<).S . .
(>.2 f 2..S,
"■4- II. 3j:
28.0-t- 5.IJ;, ;
24.fH-l2.2:i !
i
22.1 t I2.2<
l')-7- I.2i
16. I- 3.4.J ,
I
I
1848. I
12-7- 7-3-'
9-3- S.I,
7.0- r.."*, '
1.5 . •
2.0
2c) ,6
2^ . 2
23.7 . .
20.3+ 1.2,
I'l.9^ 22.5,
I
14.4+ 5-ii ;
I 1 . o -t- f) . ()n
S.5- (i.sJ
,<: -■ '■') f
Ian.
I'd,.
Mar.
April
M.ay
)unr
July
Auk.
Sept.
Sept.
Oct.
N.iv.
I)i-c.
Jan.
Mar.
.'UMil
May
iiinc
[line
July
Auk.
Sept.
Ort.
Nov.
•)cc.
IS47. '
2.(.2- 3,
:i..74 „,
20.3- 3
U<.<) .
14.4
in.S
S.4 .
5.') .
I.'■|^ 3
29.2 .
2f..7+ fi
23-3 ^
20.7 .
1S4S.
173 .
13.9- fi,
12.5- 4
' I . I — 4 ,
fi.fi- 7
3-2- 8
30.8- 2
28.3
24. ij
21.5
l').o
15.6+12
13.1+ 5
.1,'= 120 4-
Jan.
Jan
Keh.
Mar.
April
May
June
July
A UK-
Sept.
Oct
Oct.
Nov.
Dec.
Jan.
Feb.
Mar.
April
.May
Inne
July
• I Aug
■ I AUR.
. ^ Sept.
. I Oct.
8:, I Nov.
li Dec.
1847. "
1 . 2 I I .
2.S.S I 3.
25-3+ 4
24 ,) f o.
21.5' 3-
19.0+ 2.
15.4- I.
13.0
c,.(, .
f).2
3-8 ■
3t.3+ I-
27.64-11.
25.3+ o.
1848.
2r.9
15. 5- I.
I7.H- 4.
13.7- I-
II .2+ I.
7.8- o.
5.4— o.
l-O- 5-
29.5
26 . 1
23.6
20 . 2
17-7+ 7-
3: Ian.
f. r. I..
:: M.r.
1, Mar.
2, .April
8.. May
4, I line
■ J'll.v
. I A UK.
Se|)t.
. ! Oct.
ri, .Nov.
4:1 Dec.
7i r)ec.
i ,
. i Jail.
4. ' Feb.
-:. ' Mar.
5; .\pril
n, M:>y
9, I June
f., July
4. Auk.
Sept.
. iSept.
. Oct.
Nov.
O; Dec.
= 181/ 4-
1847.
2 . M
1.9.-
29.5 -
26.0 f
23.6+-
20 . o —
17.6-
14.2
10. 8
8.4
4 9
2.2
29.9 +
I84S.
26.54-
23.1-
21 . 7
18.3-
1 5 • 8 -t-
12. 1-
10. o —
6.5-
3"-7-
28.2
24. S
22.3
3-7i
3-7.
0.4,
2.71
3 ■ ',:i
O.6..
1.6,
"■7>
0.2,
I.41
I ■ -I
1.4:
o . 2 .
4 • "■ :
3->i
6.7.-
1 .0.
Ian.
I'.b.
M:ii.
.\piil
.\lJiil
May
June
July
Auk.
Sept.
Oct.
Nov.
Dec.
Dec.
Jan.
Feb.
.Mil
.\pril
.May
I line
July
Auk.
.Sl;.t.
0,1.
Nov.
Nov.
Dec.
240 -t-
1.847. '
10.4 t- 2
7"+ 5
(1.5 »- 2
3.1
30.(1
28.2— I).
21.6- 2
b, Feb.
■3,: Ma'.
Apiil
. May
3.'. J line
1 1 June
3.61
I- 2.9, ; July
AiiK.
Sept.
Oct.
Nov.
Dec.
Dec.
1S.8
15.4
13.1)
'J- 5 •
6.S
31-5
1S4S.
31. I
27.7- n
2'). 3
22.9 t 2
20.4+ 9
17.0
14.6(10
I I . I
7-7- 5
5-3+ 2
1.8 f I
29.4- 9
26 . 9
' Feb.
I
; Mar.
Mil.
A|iril
! May
Tune
July
. AiiK.
i:i Sept.
O:, Oct.
9.. ' Nov.
4:i Dec.
. Dec.
4i
300 t-
1847. "
■ ■i-o . .
-b ..!
II. I . . i
7.7 t- 3 -21
5.2-1- I.O,
I.S-t- 4.1, I
29.2-)- 1.9,
2b. 8 t- 6.8:,
23.4+ S-ljj
20.04- I Ah
17-6- 7-3.'
14. 1 o.o, [
11.4 ~ 1. 6.,
3')-i-
1848.
4-7
3 3
3"')
27-5
2.8
Ml
O.Ij
21. b t-
19.2-
15.7 )-i7-4i
12.3 + 15.7:1
9.8+ 8.3,
6.4- 4-9:1
4.1:- 7-2:i
31-5- 5.71
If we tollow any one of these vertical columns, we shall liiid that the dates corre-
sjtmid sueeessively to all points of the lunation in a |»eriod of .|I2 days. Tin; first
(d)servations of each period will he Iho last ones of the lunation, and the last ones those
made immediately aftur nt^w moon. Between <'a(di pair <il' periods will lie a gap, gen-
erally of three or lour months, during which the moon was, at the corresponding points
of mean anomaly, too near the snii to he ol»served. li the t'Wservations are etpially
scattered Ihrongh each period, all the errors arisinir troni erroneous senu-dianietor and
|)arallactic inetpialily will he eliminated. Tin; <reneral minnteiiess of these errors, and
their approatdi to a balance during each id' the periods in (piestion, are such as to render
them insignificant, if we takt; the m";in results, not hy years, Iml by periods. This is
the course adopted; the partial periods al the iieirinninLr iiiid end of the entire scries of
oh.servations being omitted. The first period actually employed was that corresponding
4 m
ii
Ismimm
20
\i
m
to tlic, soxlaiit 240 -300 ', ill wliicii llic, first ohscrvation was made 011 Jaiiiiuiy 10,
1847, and tlio last on Soj)tfinl>or 18 of (lie same year. Tiu* last jxirioil coricspomlcd
to the sextant 180-240'^', the last observation in which was on November 13, 1858.
There were, in all, ten periods corresponding to each sextant, and hence ten sets ol
(;(|uatioiis, each liiving iiu^an values of //, /i, and SI for periods extending throuiili a little
more than a year. Each residual gave an e(piation of con<lition, for th<! eoellicienls of
whi(di th(! mean value corresponding to tiie entire sextant was taken. Tiiese values for
tiie sev(!ral sextants areas follow:
1
^
sin.f
COS.!,'
sin'i'
sin^cos,f
COS«A'
1
0- 60
-(- 0.4S
+ 0.S3
0.23
+ 0.40
n.69
2
60- 120
+ O.rjC
0.00
0.91
0.00
0.0c
3
120- 180
+ 0.4S
- 0.83
0.23
- 0.40
0.69
^
180-240
- 0.4S
— 0.S3
0.23
+ 0.40
0.69
5
240 - 300
- r-../.
0.00
0.1)1
0.00
(J. 00
6
1
300 - 360
— 0.4S
+ 0.83
0.23
— 0.40
0.69
The sums of the residual errors, corresponding to each period and each sextant
arranged in chroiudctgical order, togetiicr with the number ol" residuals of which each
sum is formed, are as follow
Mean
(late.
/ = 5
1 = 6
» = I
(=2
!
= 3
» = 4
1847.8
+ 6.5
-f 14. -1*1
+ IS. An
n
+
. Oji
- if>.7i7
1848,9
■ 6.7
+ 8 .7
- 33-f^i:
- 1.931
-i
23.2i»
■f- 31-517
1S50.1
f 1.5
- 34I17
— 40 ■9*1
— 9.1^1,
+
22 . 2m
+ 33. 9»
.85.-
- 4 ■ 5 :l
- 5')-Ai-:
- 50. 7i'.,
- 23.5.^1
-
4.821
+ 20.6...,,
IS52.4
- 42.8.J
- 50.O..;l
— 48.0ir,
- 21.51S
+
35 .Oil'
+ 25.4.;,
IS53.5
- 3I-2JJ
— 106.9..,
- f)3.f'ii
+ I.2„
+
6.0.^1
- 38.>>*>
1854.6
- 3"-3i;
- 94 •'^';i
- 35 •4«
+ 4 • 2w
+
i.7h
- 24.48"
1S55.8
- 24-3n
— 30. 0|,.
- 7 •3:..
- 6.9,,,
-
22.81s
- 41. Oj.,
1856.9
- 3f'.2
- 23-8,.
+ «5-4i.
+ 4.2jf.
-
48.511
- 77.017
1858.1
- 51 •■>:•.
- 48. 9.:,
- 5f'.7ni
- 47 -Si'.'
—
7&.9«
- 46.2i„
The dales given in the left-hand column an^ those corresponding to the mean of
each liori/.ontal line.
Piil'tiiig Sj Ibr I lie nieaii value of sin ij[ corresponding to the index i, as already
given; r, for that of cos ir ; and W; for the corresponding numi>cr of oltservatimis, tiie
iiornial etpialions are:
// ,. J/+ {^ V , .V,) // -I- (2 Hi c,) k = ^^ i\
(V „,. .v,.) Jl + ^^ „ . ,.;^) /, + (^' ,, . ,s.. ,:.) /, - 2 .V, r,
(V /^. r,) Jl -f {2 », s, ci) I, -f- {:>: ;/, c-) h - 2 r. r,
The values of Ii and k thus given by the normal cfpintions formed from the system
ol residuals siiown in eac.ii iiorizoiital liin- are shown in tin! next tabh\ wliich also shows
27
tlu! way ill wlii(;li tlifv arc tn-alfd. For tin; saUt; of C()iii|tlt'lt'ii('ss, llic corrfsiioinliiifi
<|iiaiiti('n's alreaily toiiiKl for tlx- jx^'riod 1S62-74 an; added, and iiRdiidcul in tlie discus-
sion, wliicii now procet'ds a.s follows; tin; nn;lliod adophnl l»i'in<^ one wliicli, lliouirli less
rijjorons than the former one, will show in a stronger liulil the evidence on which the
new itieqnality depends.
As the basis of the discussion, we take the indepenilenl values of // and /', derived
from each series of (d)servations, which values are iriven in the second and third columns
of the tal)le. A preliminary comparison of the first series of values ( 1S47-58) with
the values of // and k derived from the formuke already ifiven indicates a din)inution of
th(! constant terms of those (piantities, so that, insl<;ad of +o".64 and + o".2.S, they
I)econie, as a first approxim .tion,
//«:r:+o".50
A-or= + o".io
These constants an- now subtracted from the values ol h and /r, leavini,' a series of
residuals <fiven in the fourtli and fifth eolunnis, which, if the peiimiic leini under in-
vestigation has no existence, shouhl I»e regardeil as due to ernirs of i)l)servation, and, in
the contrary case, slutuhl be representablc by the formula'
h' z=. — a sin (yu -f nl^ -f- accidental eriors
k zn nr cos (//-(- w/) -f- accidental errors
To show clearly how far they are thus re|tresenteil, we deteiniine a coeflicient, ix,
and an anixle, A', i»y the (;(juations
(X sin A'n — h'
acos.V::^ /r'
Tin- next two columns ifiv(! the several values of a and A' tiius ol)tained. 'I'he
nearly rciiular proi^ression of the angle iV is too striking to l)e overlooked. To st.-e how
nearly this angle can be represented as one incrtjasing uniforndy with tin,' time, \ve .solve
the iH'CCSsary eipiations of condition by least sijuares. Il is ol)vious that tin; greater the
value of a the more certain will l)e the value of N\ we tluM-efore <rive weights propor-
tional to a. Moreover, weights nearly twice, as great in proportion ar<' iriviMi to tin;
.second .series (1862-74) as containing the results from two observatories, and beinii
more carefully corrected. The values of /< and n thus obtained l)y the method of least
s{[uares are :
// — i64'.6±4 -4
H — 20 .S ± o .47
Si
m
'•A
\
an,
'31
1:^1 1
The pndiable error of a value of .\' (d' weight unity conies uiit
'i'he residuals still uutstandinif are shown in the coliiinii JS. This valui' of // is
2"" le.ss than that found from the secontl seri(!s of (dtscrvations alone, and an examination
(d'tiie residuals shows that there is a real discordaiu^e !)etween the values (d'the anu'ular
motion of .V uiven liy the two series. It is((uit(! liki-ly Ihat tiie relative weights a.«;signed
HP
ij
iii
28
to tlic (tiller scrii's of ohseiviitiDiis an- twice as great as tliev slnmlfl l)e, and that tlie
most [nobalilu value (»!' tliu angle Allies nearly halt-way Ijetwecn lh(! two values
l6r\2 + 22'\^{t— 1868.5)
and
i64''.6 + 20^.8 (/— 1868.5)
tijund from the last series alone, and Iroin the two conihined. I judge that I he most
prohaljle value is
iV= i63'^.2 + 21^.6 (/-- 1868.5),
and that the proltaltle error of the aiinuu' motion is moni than hall' a degree, hut less
than a degr( e. The eohuiiii _/'.V shows the residuals ifivon hv this value ol" N.
iMc;in
date.
1847.8 i
1543.9 I
1850. I ]
1851.2
1852.4
«853.5
1854.6
1855.8
1S5& (J
1S5S.1
I 1862.5
1863.5
1864.5
♦1865.5
' 1 866. 5
.867.5
1S6S.5
l86c).5
1S70.5
1S71.5
•S72.5
1S73.5
1S74.5
- 0.08
- 0.55
- 0.20
- U.32
+ 0.26
+ 1. 10
+ '.45
4- 0.77
f 1 . 76
- 0.17
+ 0.04
- 0.64
- 1.07
- 1.03
- 0-47
- "03
y 0-34
+ 1 . 67
+ 1.48
f l.f.5
1-2.15
I- I 1)1
+ i.y2
+ <).55
- 1. 38
- !.<)!
- I . 1)2
- 2.45
- I. S3
- 1.40
+ i>.3i
+ 1 . 82
I
4- 11.66 I
i
-f 1.23
+ 1. 78
¥ I.oi)
- 1J.15
+ O. ID
- '>-3^i
- I.4'''
- 1.5''
- 1.14
- 0.3^
- o. 12
1^ 0.16
H 0.60
- 0.5S
- 1.05
- o . 70
- 0.82
- 0.24
4- 0.60
+ "-OS j
+ "-27 I
'h 1.26 !
- 0.67 !
- 0.46 I
- I. 14 I
- 1-57 i
- 1-53 !
- i>.<)7 j
- 1-43
0.16
)- 1. 17 I
+ o..,S I
I 1. 15
h 1.65 :
I 1.41
( 1.42
+ 0.45
— I.4S
— 2.01 i
I
— 2.02
— 2-55 ^
— 1.98 I
— 1.50 j
+ 0.21 I
+ 1-72 {
+ 0,50 I
!
+ 1.13
+ 1.63
+ <'-'W j
— 0.25
0.00
— 0.46
— 1.56
— 1.66
— 1.24
— 0.46
— 0.22
+ 0.06
+ 0.50
a
1
0.74
52
1.82
145
2.13
i6l
2.13
15S
2.56
175
2.07
>97
1.77
212
0.34
30S
2.13
328
O.S3
50
1 .22
22
2.03
34
1.35
58 !
».55
99
0.97
<JO
1.50
108
1-57
174
2.03
215
1.58
2IS
1.24
248
1.66
262
1. 41
272
1.50
289
I
3
3
3
4
3
3
h
3
t
3
5
5
4
2
4 I
4!
SJ
5
3
I
:!
/I + Itt AN
94
118
141
165
IS.)
212
236
260
284
3<J7
40
61
81
102
123
144
165
1S5
206
227
24S
269
289
+ 42
- 27
- 20
+ 7
+ 14 I
+ 15
+ 24
- 4»
- 44
- I1J3
+ 18 i
+ 27'
+ 23
+ 3 ,
+ 33 I
+ 3(>
- 9
A
N
4
24
44
36
-
H
u
+
2
-f
12
- (<t>
- 55
- 112
12
21
19
1
3"
34
II
: - 3u
- 30 ;
1 — 12
- 12 t
— 21
— 21)
- 14
— 12
- 3
— I
"
^ -i
'I'he old and new series ol ohservalions agree well in giving lor the value ol the
eoeHieient ol' lliis li'rni.
'I'iie old series, rr — i".66
'{"he new series, n r:z l".55
The (died ol' the accidental errors will lie, on the whole, to increase the value ol'
the coellicieMf. I consider thereliire that tht^ value
<f — I .50
29
iiiiiy !)(■ iiiliiptrd iis llu! luosl prohabli! wliicli ciiii l>c derivotl from all llu; ubsorvatioiis.
11' we siilitiiicl, IVoui fiicli value of // aiul k in the preceding table, tlio perindic
portions
//zr- i".50>iH [163". 2 + 21^.6(^-1868.5)]
/,'= i".50(:os [163^.2 + 2r'.6(^- 1868.5)]
\\\n\ liikc llii' iiu'iui viilue of llie outstanding remainder for (!uch scries of oltservalions
we lind it to Ix; as follows :
Old scries, //o = + o".33; /-o = — o".i7
New scries, h^ — -\- o".65 ; A-q = + o".36
Till! (liUcnMici's, o".oi and o".o8, between these last values and tliost; found on page
23 arise from the dillcrcnl value of the periodic term. I consider that the results of the
second sciics arc entitled to three times the weight of thonc of the first, and shall there-
tiirc put l()r the dctinitive values of h and k,
//=z + o".57 + /''
Z— + o".23 + A'
The corresponding corrections to the eccentricity and longitude of perigee arc:
Sf- — 0".2() X
r67r — -\-o".\2 /
<5;r — +2".2
Tlie corrcclions to the moon's longitude are: /
'V n — // sin ij' — /-cosi' /
r: — o".57 sin i,' — o".23 cosir+ i".50sin {g + N —c)o'^).
Tlif last term istht; liithcrto-uiisus[)ected ineipiality indicated by observations, but not
vet known to be given by theory. It may be either an inequality of the ecointricity and
perigee having a period cd' about \(i% years, or one of the moon's mean longitude having
a period of
2 7''.4304 ± o''.004o
Substituting first fi)r A', and then fori,', their values in terms of the time, the expres-
sion fi)r the inciinalitv of longitude becomes
I w < *
i".50 sin [- + 73^2 + 2i'^.6 (/ - 1868.5)] = 1 ".50 sin (56^.8 + 13^.12413 0,
7- being tiie tinn; in days (;ounted from fjrceinvich mean noon of 1H50, Jan. o.
It, nould pcriiaps l)t! premature to introduce so purely (unpirical a term as this
into lunar tables for p<'rmanent use; but where, as at present, it is recptisite to obtain the
cnrrcelioMs In the tal)l('s (hiring a limited period with all possible accuracy, tht; eviih'nce
in IJivor of the it ality of th(! term seems strong enough to justify its introduction. The
niilv .ippiirciil cause to wliicii the term can i)e attributed is the attraction cd" sonic one
of llic planets.
In the investinalioii (d' corrections to the longitu<le, it only remains to determine
tlie slowly-vaiyiiig conci'tions to the mean longitude, or to « f5r, given by the observa-
tions. To defeiiiiine tlu' errors of short period, we have applied several corrections to
I lie residuals, not as real, but only to render the various ol)servations comparable. We
80
liavc ii«»\v to con.sidur (lit; jMin; results of obscivatiuiis as lliey would liavi; l>i!eii liad tlicso
corioctioiis not bt;cu applied. Tliesc lor tlio second series of observalioiis are loiirid
I »y taking tin; sum 01(1) the incjan of tlie small corrections, ap[)!ied on account of
(>')S(!rvatjry and limb, to compensate lor the systematic dillerenees l)etvve('U results from
dillerent lind>s or dilferent observatories; (2) general corrections to make the residuals
in the mean very small ; (3) remaining outstanding corrciction found by solving the
cipiations of condition.
The corrections from l)oth series are as follow: the corrections sinc(^ 1862 may
be very closely represented by a term increasing uniformly with the tiuK!, as is shown
l)y the last two tolumus.
First scries.
Date.
u>h
Dalu.
IS53-5
II, h
18.7.8
- 0.15
••
+ 1-77
1S48..J
- 0.43
iS54,f,
+ > • 40 1
1850.1
+ 0.32
1855.8
+ 1-24 1
1S51.2
+ 1. 13
iS50.()
+ 1.50
1852.4
+ 093
185S.1
+ 2.40
.■.;ti
Second series.
Year.
(I)
+ <'-45
(2)
+ 2.10
(3)
t 0.04
II ih
a-\-l>l
+
A
1.07
1S62.5
+ 2.5'J
+ 1.52
1S63.5
+ 0.45
+ 1.2l>
— 0.27
T- I. 38
+ 0.60
+
0.78
1.S64.5
0.00
0.00
— ".49
- 0.4IJ
— 0.32
-
0.17
1S65.5
- 0.15
- ■•>5
- 0.62
— I . 1J2
- 1-24
-
0.68
1866.5
- 0.15
— 2.m)
- 0-75
— 2. (JO
— 2.16
-
0.74
1S67.5
- 0.15
- 3-40
— 0.41
- 3')6
— 3.oS
-
0.88
1 868. 5
- 0.15
- 4.05
— 0.20
- 4.40
— 4.00
-
0.40
lS6(j.5
+ o.oS
- 4.85
— 0.2I
- 4.')8
- 4-92
-
0.06
1S70.5
+ 0.08
- 5.50
— o.o()
- 5-51
- 5.84
+
0.33
1871. 5
0.00
- f'-35
- 0.52
- 6.87
- 6.76
-
0. II
1872.5
— 0. 15
- 7.25
— 0.22
— 7.62
- 7.68
+
0.06
1873.5
o.ou
— 8.30
+ O.IO
- 8.20
- 8.60
4-
0.40
1S74.5
0.00
- 9-45
+ 0.38
- 9-07
- 9- 52
4-
0.45
§2.
INVKSTICAIIOV OF THE TOLAR DlSTAIvICK AN'D l.ATirrDK.
It, is a siiiLnilar «-irrii!ii.staiic(' lliat diiriiig the lust six years, at least, the (li)si'rva.
linns of tlic niooa's |ii»lar <listaiu'(! are imicli lt>ss accurate than those di' its riulit ascen-
sion. Wlietlier this is to he altrilinted to lh»i instrnnients, or wlietiier it is a resiiH ot'
f^rent irreirnlarities in the oiitrnie of the; lunar glol)e in tiie polar reitions, cannot at pres-
(Mil i)e (lecitled. To whatever cause wo atlrihutt; the errors, tiieir existence renders a
rit'orous treatment ot the in;lividual observations of little value. We shall therefore,
from the wiiole of the errors in dediniition, sook to olttain the i>est corrections to the
inclination and node of the moon's orl»it.
Fvom the derivatives of the moon's d(;clinatiou relatively to its true lonizitinle, the
inclination, and the node, whieii have already boon given, wc oi)tain:
ot 4-
dl ^ do
„ » no 5, no „^ , (Id „
no zr ''' + o" + ,. "'
dS ..
di
ly heing known tVoni tin; data already givcMi, the (Hjuations ol condition will be
thrown into the li>rm
d''^ ■ ^r^ . dS ^. 5. ~ dS f,
r , I OO -\- -— 01 — 00 — ,, 0/
I do ' di di
Vrom llie numerical cxpressitnis already givcin, we have
-'''^ <S/ — sec <^ [(0.40 + o.oS cos 0) cos /+ o.oS sin sin /] rV
dl
If we put
fSA =:tlie (•orre<:ti(Hi to the moon's mean longitude,
K ziz 0.40 -\- 0.08 cos 0,
II =:0.oS sin 0,
wo iiave tlie quantities of the first order, with respect to tiio eccentricities,
— [A'cos / -f //"sin /] [i +2 r cos (A — ;r) ] see rS
The largest terms in sec S are
1.040 -f- .016 cos ^ — .040 COS 2 A — .016 cos (2 A — 0),
while, it" we replace / by the mean longitude. A, we shall have:
/=r A -|- 2 c sin (A— tt)
sin / =: sin A + '' ^^iu (2 A — w) — v sin tt
cos / zr cos X -{- f cos (2 A — tt) — t' eos tt
I s
If we sid>stitn(»! these various (luantitie.s in the expression tor ,, 6/ \\(> shall lind
tlf
mmmum
iPf*'i
M-
■MMH
3^
no sciisihlo tciins (Icpciidingon tlio sine or cosine olllio argmncnl of liililiulc, A — 0.
we siihslitiile lor SI its vulut! in <5A, wo sliiill iind tlu; priiKiijial Icriiis in cos (!•
,// </\
to
1)0
A' COS A + /Tsin A + 3 « /vcos (2 A — ;r) + 3 c //sin ( :; A — /t)
-f 0.9 cos / — o.: sin /
III consoqucnco of the great number of revolutions ol' \\u\ moon tlirougli wiiicli tin;
ol»s(!rvations now umler discussion extend, T have considered tlml all (!xcc|)t the first two
l(!rms might he treated as accidental errors, which would cancel each oilier during the
course ol' tlu; oliservatious. Using for S\ the iiutan correclions lo liie moon's loniiididc,
we have the lollowiug values of the correction to the dcclinalion for Hiosi; errors of
h>iigitude:
Year, CoiTi'ction.
1862.
1S63,
1864,
I 865,
1 866,
1867,
1868,
1 869,
1870.
. 1S71,
1872,
1873.
1874,
The mean correction to the moon's tabular north-polar distance for ciich year, from
observati(ms of each limb at each observatory, was taken with a view of detec-tiiig any
constant error of sufficient magnitude to alfect the final results for errors of liie iioth^
and inclination. These means should have been taken aller the application of the cor-
rections just found: actually, however, they are tlic mean corrections given by the
observations, allcr applying the following constant corrections to reduce tiie deilinalions
to tlic same fundamental standard :
+ 0.6
— 0.1
— O.I
0.0
-0.6
-f 0.1
-0.8
00
— 0.1
— 0.1
- '4
— 0.2
- 1.8
-03
2.2
-0.4
- 2.8
— 0.6
-0.6
-3-8
-0.5
-4.2
— 0.4
To Oreenwicli obacrvationB of N. P. D.
To W.isliington observivtiniis of X. P. I).
1862-67, —0.4
1868-74, -f 0.2
1862-65, -}■ O.^
1866-67, — I.I
1868, - I 2
1869, —0.6
(870-72, —0.4
•873-74- 0.0
The.SC corrections are approximately those necessary to reduce flic star-observa-
tions of the several years to Auwers's standard of declination. The change in the Green-
wich correction between 1867 and 1868 probably arises from tin; introduction of a new
33
constiuit of refraction in 1868, \vliil(! tlio cliange in tlie Washington corrrclion in 1866
corresponds to the introduction of tlio largo transit circle in place of the old mural circle.
Year.
Correction to N. P. D. given by —
Greenwich.
Washington.
N. L.
S. L.
N. '..
S. L.
1862
ti
— 0.1
- 0.8
II
- 0.3
- 0.8
18&3
+ 0.2
- 0.9
- 0.5
— I.I
1864
+ 0.4
- 0.6
+ 0.8
- 0.9
1865
+ 0.5
— 0.2
+ 1.2
— 0.2
1SC6
- 0.7
- 0.3
+ 1.4
- 0.6
1867
- 0.4
- 0.6
+ 0.1
— I.I
1 868
- 0.7
— I.O
+ 0.2
+ 0.2
i86g
— O.l
- 0.6
- o.S
- 1.7
1870
- 0.6
— O.I
— 0.1
- 1.8
1871
— 0.2
- o.B
+ 2.1
- 1.8
1872
0.0
0.0
- 0.7
- o.S
"873
- 0.9
+ 0.1
-f- 2.0
— 0.1
1874
• •
• •
- 1-7
-0.5
The large residnals of tl.c Washington observations of the south limb led to the
application of the farther syslematic correction ot + i"-0 to all those observations before
conibining them all. The corrections arising from the error of mean longitude were
then ai)plie(l, and the out.standing residnals were considered to arise from accidental
errors and from errors of the inclination and node. The equations of condition thus
betiomc
0.92 sec S [sin (/ — 6) 6i — cos {I — 0) i 69] = dS
or
sin (/ — 6) Si — cos (/ — 0) i SO = 1.09 cos SX^S
Owing to the smallness of the final residuals, 66, the factor 1.09 cos 6 may be consid-
ered as a constant, and, in the actual solution, has been put equal to unity. lis mean
value is more exactly 1.04, and its ellect may be obtained by dividing the final results
by this factor.
The final values of the residuals were then arranged according to the values of
X— 9,or the moon's mean argument of latitude, as the residuals in right ascension were
arranged according to the mean anomaly. The sum of the residuals corresponding to
each interval of 20'^ in the argument, with the corresponding number of observations
for each year, is shov/n in the following table :
5 M
n^A-'Twmsmisfm^
I
J
II
34
Slims of errors of the moon''s eorrected dedinafiou, f/irrn />// ohserrntious at fireetnrieh
and Wiishiiif/toii.
LimliSdf ?.
iSf,
2.
1S6:
2.M
N.
186^
•
186;
i86(:
.
1867.
X<J.I N.
1868.
£i1.t
N.
1S3
N.
2.M
N.
£.!<)
N.
SiM
N.
o to 20
- 3.3
8
+ 1.3
3
It
+ 4-0
8
+ 5-4
9
+ 26.7
II
- 2.5
8
+ 0.4
9
20 to 40
+ 9'f'
9
+ 5.8
7
- 0.4
9
+ Co
7
+ 2.6
12
- 2.3
9
- 6.1
•7
40 to 60
- 1-4
9
+ 6.9
10
+ 6.5
6
+ 9-7
7
+ 4-0
9
- 4.9
10
- 7.9
«5
60 10 80
0.0
7
+ 16.4
10
+ 6.6
8
+ 8.7
12
— I.I
16
+ 14.5
II
— ".J
5
80 to 100
4- 8.6
II
+ 0.4
12
+ 11. 1
6
+ 7.9
II
+ 5.5
7
+ 0.5
10
-1J.8
12
100 to 1 20
+ 3-2
7
+ 8.5
15
+ 3.2
5
+ 7-4
7
— 1.0
8
- O.I
6
+ 0.2
6
:2o to 140
-',.2
12
+ 3.1
8
- 6.1
S
+ o.S
II
+ 11. 9
14
-12.4
8
- 6.8
II
140 to if)0
- 0.3
4
- 4.6
9
— 2.2
5
- 9-7
15
- 1.2
10
- 7.7
12
+ 6.2
14
160 to 180
+ 0.5
9
-10.4
6
-10.4
12
+ 0.5
9
+ 2.2
10
- 8.9
9
— n.2
9
180 to 300
- 8.6
6
- 5-7
II
- 0.6
7
- 5.3
13
- 7-7
6
— 15.2
14
+ 3.1
10
2(XJ to 220
-22.3
8
— II. 6
10
+ 4.7
12
- 5.4
9
- 3-3
ID
- 6.8
14
-II. 8
II
220 to 240
-14.4
12
— 10.2
9
- 8.8
10
— I.O
7
— 2.0
'3
- 5.9
12
— 10.0
13
240 to 260
-12.4
7
-12.3
9
-4..
8
+ 4.6
II
+ 1.2
9
- 0.6
9
- 9-2
15
260 to 280
-2.3
4
- 32
4
- 8.5
8
+ 1.5
9
- 5-3
9
+ I.Q
8
+ 1.9
9
2S0 to 300
— 2.S
7
- 4.3
8
- 8.4
II
- 4.0
4
- 3.5
■>
-II. 4
J3
0.0
«3
300 to 320
- 7.1
10
- 6.2
10
4- 9.6
8
+ 3.1
5
— 0.1
13
- 8.4
10
+ 0.4
8
320 to 340
+ 2.0
7
+ 3-4
8
+ 6.0
12
+ 8.6
6
+ 8.4
II
+ 2.1
<>
- 6.7
14
340 to 360
+ 7.3
5
~ 6.5
5
+ 4.9
13
+ 11. 6
8
+ 7.0
14
+ 2.9
3
- 3-1
12
—84.0
142
-75-0
J54
-49-5
156
—25.4
159
— 25.2
191
-93.1
'75
-85.9
203
+31.2
+45.8
+ 56.6
+ 75.8
+ 69.5
+44.3
+ 21.0
+ 12.2
-52.8
—29.2
+ 7.1
+ 50.4
-72.1
-73.7
■iMM
wimtm
85
Sums of vrrnrs of tin' iimotCs ainrctrd (Icclhiutioii, d'c. — C(»iitimic(l
1873.
1869.
1870.
1871.
187a,
Limits ufX,
1874.
o to 30
20 (O 40
40 Id 60
60 tu 80
80 to 100
100 to 120
120 to 140
HO to 160
160 to 180
I So to 200
200 to 220
320 to 340
240 to 260
260 to 280
SSo to 300
300 to 320
320 to 340
340 to 360
IMJ
II
+ 7.1
+ II. a
+ 6.4
- 5.0
! - 2.0
- «3.7
- It. 4
i - 15.4
- 2.5
- 5.4
- 5.4
- 6.6
- 18.4
- 7.7
- 11.4
+ 5.3
+ 5.7
0.0
-104.9
+ 35.7
— 69.2
7
9
II
7
9
12
II
9
4
6
7
13
7
5
10
+ 3-7
+ 6.6
+ 8.6
+ 3.5
+ 6.2
— 6.2
+ 4.5
-II. 7
— 5.7
— 0.5
— 10.2
— I.I
— II. I
-15-4
— 10. 1
— 9.'
-10.3
— 1.2
><.
ill,!
7
- 3.8
10
— 0. 1
10
- 0.8
7
+ 13.2
9
+ 13. 1
12
- 6.3
7
- 1.9
II
- 4.6
13
- 5.1
6
+ 5.4
12
- 6.2
9
+ 9.1
S
+ 5-6
15
- 6.2
5
+ 3.6
7
+ 3-8
12
- 5.8
6
- 6-3
4
II
II
9
9
8
7
9
II
10
•4
8
7
8
8
9
5
Sil.S
N.
:;.i,i
n
It
- S.o
C
+ 9.0
- 7.0
6
- 7.7
— 1.2
10
- 2.8
- 5-1
7
-'3.7
- 3.7
8
+ 4.8
— 2.0
8
— I.O
+ 0.2
14
— 2.2
- 8.9
9
+ 7.4
- 4.6
12
- 3.9
- Co
3
- 8.6
- 2.9
9
+ 5.2
- 4.5
10
- 2-7
+ 13.7
13
+ 14.8
+ 2.2
II
+20.7
+ 3-8
9
+ 3.3
- 1.3
II
+ 4.3
— 12.2
12
+ 14.0
+ 0.3
9
+ 7.2
155
-92.6
+ 33.*
166
-47.1
+ 53.8
-59.5
•1- 6.7
153
-67.4
-47.2
167 ; —42.6
+90.7
+48.1
It
9
+
7.7
13
10
-
7.5
7
-
17."
11
'
-
25.4
14
-
5.3
6
-
22.4
12
4
+
2.1
6
-
12.6
II
-
5-9
10
-
7-2
3
-
'9.3
12
-
15.2
6
12
-
4.6
6
9
-
3.5
6
10
-
5-5
13
S
+
0.3
lo
7
+
4.0
10
II
-
3-9
9
169
40
-
154.3
+
14. 1
140.2
The goiit'fiil irregiihu-ity of the residuals in (lecliiiiiliou is such that no great ad
tage woulil result in a separate solution ot" the equations for the separate yetirs.
sum of the residuals for each 20^ of the argument was therefore taken during the w
thirteen years of observation, with the following result:
van-
hole
x-e
iAr!
N.
A-O
SAct
N.
„
,/
to 20
+ 47-2
103
I So to 200
- 62.3
no
20 to 40
+ 10.7
115
200 to 220
- 95-3
12S
40 to 60
+ 6.1
119
220 to 240
- 73.3
126
60 to 80
+ 12.3
124
240 to 260
- 32. s
127
So to 100
+ 34.3
121
260 to 280
— 23.8
106
lOO to I20
— 3f'.2
III
2S0 to 300
- 50.1
123
120 to 140
- 27.4
120
300 to 320
- 5-4
115
140 to 160
- 65.3
124
320 to 340
+ 19.2
122
160 to 180
- 65.4
126
340 to 360
+ 20.2
no
.1
^.m^f^fiKm^.- imt M iUi
-~^—- ■—^r~——^,i-Li^:±jMiiM^At,
|w|
Ti<'uviiij( in tin; ('(|iiiilinii« a ntiiHlni\l, Irrni '"i/i, n'j»n'si'iitiiiff (lif inniii roiisluiil crinr
still oiilstatiiliii^ ill tlic iiicasiircs of tlt'diiialiDii, tin; solution of tin; ('(|iiatioiis of coii-
<litioii given \>y tin; residuals gives the l()llowing results:
Jl>--n".i7
Ji, = + o". I 5
iJO- — iy"..\o
or.
Correetioii to the iiielination, — o".i5
Curr(!elioii to the longitude of node, +4". 5
This correction to tlie longitude of the node from Hansen's tabl(!S iniplies a diiiii-
luition of the seeiilar retrograde; motion of the node, which is (|uitr; aecor<lant with the
results derived from ancient eclipses. Hansen remarks that an increase of 12" per cen-
tury in the longitiuh; of tli<; moon's node will improve the agreeni(;nt of his tables with
ancient eclipses;* and, if wesui»pose the tubular longitude of the node to have l)e(;n (;or-
reet in 1825, this would imply a correction of-f5".2 to the longitude; of tlu; node
in 1868.
' l)iirli';;iiiijj, t'tL'.,Tli. il, p. y)l.
If i
^^ W(
4 ,
37
HH
Al'Ml.lAKV lAII.KS R)K !• \( I I.I I A 11 N ( ; rill', (( )M I'l I \ IK ).\ oh llll, CoKkir
•I'lONSTO IIANSKN'S '-rAIILKS Dl', I, A I.INK ", (ilVKN \\\ IIIK I'K lai I >l NC DIS
CUSSION.
Tlio loUowiiiif is a siiintuiirv of Ww, ('(trrccrKtiis to (lie l(iii^itii(l(! id' llic inooii riom
IIiiiiscm's lul)lcs 1,'ivcti by llic preceding disciissioii. I'iie first six leniis an; iippliciililt!
(() tli(! (lisliirhed mean Ningilnde, or '^'■Ari^itmen/ foiifliimr/i/ii/"; Ihe remainder lo Hie
(rue loiigididi!; but tiic}' may all lie used as correclions of the "Ji^mnr/il foudaiiiinlnl"
widioiit serious error:
Concclioiis on accuitiil of (liiiiinution of the sular pdntl/nx . . n <5c z= + o" .(.)6 sin I)
-\-o".o7m\{l)-g)
On (icrount of /iif:r)//u'xis [lure in'orisioiHillij sef oxldr), that
the moons atilcr of i^rorily t/oc^ not coincide with the
center of figure, together with the correction to the erec-
tion resulting from llie correction lo the eccenlr'iciti/ . . . n i^z rz + o".09 sin g'
— o".3,^ sin : I)
— o".2i sill (2 I) — ii)
On iiccoiiiil of term nceidenlallu iiitroiliieed into
the tiihles with a wrong sign ('ir — — o".62 sin {2 g — 4 g' -{- 2 m — 4 <>>')
On account of correction to the eccentriciti/ ond perigee
found from observations during i S47-74 6r zz — o".57 sin g — o".23 cos g
=z o".62 sin (g'-f 202'^.o)
Empirical term, neceamri/ to satisfy ohserratio)/s,
hut not verified by theory + i"-5" '*'" [a' + 2 1 .6 ( >'— i S65. i ) J
Unexplained correction to the mean longitude, changing slowly from year to
year See Table IV.
The deduction of all these terms, except th(! last, has been fully given in the pre-
ceding pages. This secular correction to the mean longitude has been derived from the
outstanding errors of mean longitude given on pag(! 30, in the C(dumn n Sz, l»y suppos-
ing this quantity to vary according to some simple law, which law changes Avhen necessary,
so as to satisfy the observations witliin the mean limits of their probable error. An
examination of Table IV sliows, that, from 1848.0 to 1855.5, t''*' eornjctiou is supposed
to increase uniformly at the rate of o".20 per annum. It is then supposed to remain
constant until nearly 1S63.0, a period during which the observations are not continuous,
there being no comparisons with theory from 1859 to 1861 inclusive. From 1863.0
until the present time, the observations are well represented by the corrcctioii
— 5"-53 — o"-S6(^— 1870.0) -fo".02 (^-1870.0)-
The continuance of this correction beyond 1875.0 is, of course, purely conjectural.
TAr5I,ES FOR APPLYING TIIK PRIXKDIXG CORRECTIONS.
The following tables are designed to facilitate the computation of the corrections
mmm
m
.■■}
M
38
just <.'ivfi). To avoiil tlic iicccs.sity of lelcniii^ to Huiiscir.s (aides, the valiU's olall the
necessary argimieiils arc yiveii lor the years 1850 to 18S9 in 'lalth's I to 111.
Talile I: tlie epochs are January o, Greenwich mean noon of common years, and
January i »d' leap years. All the ar^fumeids increase iiniforndy i)y a unit in a day.
Ar^rnment g is the moon's mean anomaly, converted into days hy dividing its ex-
pression in deirrees l)y 13.065. It is equal to llan.sen's argument g diminished l»y 15 days.
Argument D shows the number of days since mean new moon, or, it is the mean
departure of the moon from the sun expressed in days. It is ecpial to Hansen's argu-
ment ^^ diminished by 30 days, or, which amounts to the same thing, by o''.47.
Arirument A gives the number of davs from the time when the anjjle
2g — 4g' + 2a) — ia)'
was la.st zero.
Arirnmeiit li is that of the empirical term indicated by observations, but not given
by theory.
Ar<rnment u is that of latitude, or the number of days since the mean moon last
passe«l her ascending no>l('.
Tiddes II and III do not seem (o U(.'ed explanailoii. In using the former, ean; must
l»e taken 1<i diminish by ou(! day tiie dates for Jauu.iry and ^"ebruary of leap years.
Talde IV gives tht; secular corrections to the mean longitude, or to n6~, obtained
from ol)servatioiis in the manner aln^ady described.
Table V, argument J, gives the correction for the t(.'rm introduced into the tables
lescribcd on paye o. I
witli a wrong sign,
tilde, and is therefore designated as 6v.
properly
ippli
Talde \l irives the empirical term, whis'li, so far as is known, may be a)t|ilied to the
true loiiiritude.
Talde VII "fives the sum of the terms of mean hjngitude
+ o".96 sin 1)
o
■OJ
Sin
I)
-o".i3sin(7> + ir')
4-o".09 sin g'
Tl
le sun's mean anomaly, g', having a ])erio(
dof
I year, the sum of these terms can
lie expressed as a function ol' .'> and llie nioidli, and is given in the table for the middle
of each monili, and for each day of J).
Table VIII gives (h(! sum of (he terms ol true longKnde which depend wholly or
partly on the moon's mean anomaly, namely:
+ o".62 sin (g -f 202".o)
+ o".o7.sin(/; — 5-)
— o".2i sin (2 D — g)
Tin- sun; of the terms of n S: are to bo reduced to eorreclions ol the 'ongitude in
orbit by multiplication by (he fad or
1 -f- 2 f cos r -f -^ c'- cos 2 g.
This factor, less nnitv, is iriven in T.ible IX.
!^
39
'or
convenience, the unit of the faclor is uniittctl from the lahiilar luiiiil
necessary to adt
icrs, so llial
it IS only necessary to add the product / X « '^- i" ""i<h n S: and Sr to hav(> tiic cor-
rection of the true longi(nd<' in orlnt.
These corrections being applied to llie longitude of the moon's center from Han-
sen's taldes, that longitude may he regarded as correct, exce[)ting a small correction,
wiiicli may probably be regarded as constant during any one period not exceeding six
months, and which may be attributed to tiie adoi)ted position of the e(piinox. It will
l)e best determined from occultations of stars observed at points whos(! longitudes from
Greenwich are accurately known by tehigraph, and will then be applicaljle to the
determination of the longitude of any station from occultations.
If the corrections here deduced are applied to the (,'rrors of the lunar eplunneris
derived from meridian oliservations, it must bo remend)ered that thesi; observations are
made on the moon's limb, while the corrections are applicable to the center. Hence,
the value of the moon's semi-diameter must, if great refinement is aimed at, be varied
with the ob.servef, the instrument, and tin; brightness of the sky. For large instru-
ments, Hansen's semi-diameter is about i" too great, even at night.
The sum of all the terms of n S:, Sr, and FX" ''^- f'''>"^ the tables will l)e the
correction of the longitude in orl)it. Tliis will not be rigorously the same as the correc-
tion to the ecliptic longitude.
Table X gives the small factor (F. I) liy which the orl)it longitude must Ijc increased
or diminished to ol)tain the ecliptic lonirituile. This tiictor may g(.'nerally be disregarded.
Table X al.so gi\es the data tor the- correction of the moon's latitude, namely, (i)
a flictor (i\ /?) by which the correction of the moon's argument of latitude must be
multii»lied; and (2) the term
•^/^ = — o". I 5 sin u
arising from tlie correction to the t.djulav ni,-lin.uion of <lic moon's orliit. The correc-
tion of the moon's arguaient of latitude being that ol' her lon:;itu(le, miinis the correction
of her node, the wlude correction to the latitude will l)e
dft-dft, + (F.ft) {6l--^".s-)
Table XI gives the factors for converliuL' correction'-' of loniritude and latitude info
corrections of right ascension and declinatio!:. The nn ninhv are
S.Al — or + (f . a) Si- + (/? . a) 6/i
S . Dec. =: S/i + (/• . S) Sr f (/? . S) Sft
The side argument is the moon's longitude, auil in the coefHcients iv . n) and perhaps
(/y .«) regard must be had to the moon's latitude also Tlirci; columns are therefore
given for latitude,
and + 5 respectively.
mmm^
40
-'■ I
As an example of the use of the tables of corrections, we will commence the
deteniiinatiou of the corrections for Scsptembcr, 1S74. AVe find the values of the argu-
ments for September i, from Tables I to III, a.s tbllows:
g
D
A
li u
1
1874 . . .
Sept. I . .
I'eriDtls . .
Arg. Sept. I .
A TR. Oct. I .
5..»
23.6
-27.6
12. t
7.8
S.o
1.9
20.0
24.6
-27.4
1.9
26.2 i
— 27.2 '
1.4
19.9 9.9 1 17.2
0.9 1
3.3
20.3
1 ; 1
39-9^! 47-2)' 30.0/:
or 7.8 i^ or 19.8 ) or 3.7 ) !
1 1
rig.g
: 1.4
D-j;-.
-18.5
The tabular numbers are then found as follows, with an argument incrc
unify each day. From Table VIII, we take a mt-an from columns i8 and 19.
a.-<ing by
September
Ta
lilc 1V(h,1
VI (,lr.
VII (.(
fc).
VIII (
r) .
n<
:XA,Tal,l
■IX
6v
. . .
ih'
-4".5 ■
Ta
lili: X (F .
/J) .
(■'■
-4".5)(1^
/^).
•Vi
. .
I — 9. II
I + 0.40
— 1.07
— 1.29
— o,f)5
- 9. 1 1 - 9.11
i
+0.55 +0.62
- 1 .29 — 1 .42
- 1.25 - I. 12
0.72
0.77
- 9. II
+ o 59
- 1.50
- 0,95
- 0.7S
— 9 12 —9.12
+ 0.4S ' (- 0.28
—1.48 1 — 1.40
— 0.76 I — 0.50
0.75
o.f)9
— 9. 12
+ ti.05
— 1.23
— 0-37
— 0.60
-11.72
-II. 82
- 0-93
- 1 1 . 80
■11.75
— 11.6J , —11.49
0.71 , - 0.47
-12. 84
-17.3
-+- 0.088
- 0.03
- 1.52
J's longitude
(i +(».'>)) i!f ,
(,3.«).V . .
AM . . .
(v . il) M' .
(I ■<-/)..)) iVi
iIDcc. , ,
1-55
46.5
-i3.o8
-12-75
^17.2
-t- 0.082
— 0.07
- 1.3S
— 12.51 —12.22
-17.0
-16.7
+ 0.070 + 0.056
O. 10 : — O. 12
1. 19
0.93
0.28
0.06
— 11.27
-1- o.i3
— 11.91
- IC.4
-I- 0.038
- 0.14
- 0.63
-11-55
— 11.09
- I--I5
Co. 7
13.49
-+- 0.47 I + 0.32
— 1.29
74.6
83.1
-13.84 j -13.81
-1-0.16 ! -f 0.02
0.77
101.5
—16.0 —15.6
-f 0.019 ~ 0.002
- 0.15
— 0.30
0.15
o , 03
9.12
0.18
I .0!
0.22
0.48
+ "-37
-10.64
-15-I
— 0.14
+ 0.33
- 0.45
114-5
-13-
-12.6
-0".84
- 3-70
- 1.50
- 5-2
-13-2
-o'.88
— 2.64
- 1.42
4.1
-13-7
— o". 91
- 1. 41
- 1.23
2.7
•13-8
-0-.92
- o,l3
n-5
.90
— I2.6r 1 —
— 0.09
-12.7
— o'.85
-t- 1.02 -(-2.05
- 0.77 - 0.44
+ 0.2
-j- 1.6
+ 3-5
^
•tmm S*:s . sS3KS S S
■^
41
This (..miMifalion Ims Ik.-.;,, cm.fiu,.,,-.! tu .875, Jai,u,-,-y 3 ., an.! ll.r n..s,.lls ;nr
slKtwii III ilic lollowiiiy tal)Ie:
Corrections to tlw Ephrnirris drrircl from Jfansnfs Tables of the ^foo^, fhr firrninhh
inmn ,»,.>„ of cuh itnij, from 1S74, Srpfemhrr 1, to \^--^, .htnnor], 31.
D.itc.
Corrociion to latnil.ir —
Date.
Coircciicui lo i.ij.iihir-
GM
Gr
. nicm
~
—
I
-T
^-
n
oon.
^:-i.
Lonif.
Lat.
i R.A
Dec.
tir. nu-; ii i
no„n. I ■-on,^^
Lat. R. .A.
Dec.
■ .
^^
lS7(. '
St
r'-
— I2.S
— I.
<■ —12,
<> - ■;.:
(JlI. II — 7.5
+ '.I - '>.1 -r 3.6
2 12^
I .
'- '3-
2 4.1
>2 7.2
1.0 6.S ^ 3.2
3
12 5
'•
3 13.
7 2.7
13 ('.')
I.O 6.S ' 2.7
1
A
12.2
I.
' >3.
^ — 1.2
14 i 6.6
"•8 : 7.0 2..,
!
5
II. 9
0.
5 ; 13.
5 + 0.2
'5 j f).4
^•7 7.0 1.3
1
t
-W.I,
— 0.
5 j —12.
7 4- 1.6
If) i — 6.2
+ 0-5 - 7.0 + 0.6
7
II. I
— 0.
I II. 6 2 7
17
fi.i
0.3 6.9 ' - 0.2
3
10.6
+ 0.
: i 10. <
> , 3-5
18
6.2
! + 0.1 6.S I.O
9
10. 1
0.
> i 9.<
>\ 4.1
'9
6.4
f — "•' 6.6 i.s
r
i
10
.J.6
o.-
j %.i
' i 4-3
1
20
1 „
6.8
; 0.4 6.6 2.5
II
- 9.0
+ C.f
) - -■'
) ' + 4.4
- 7-5
— 6
— f'-O 1 — 3-3
12
5-3
I.'
7 • '
4.2
22 j 3.3
1 0.9
7-3 ', 4-1
'3
7.'j
1. 1
''■:
3.S
23 9.3
I . t
■^•i 4.7
M
6.3
I.I
6.2
3-3
24 ! I". 4
'■3
9-2 5.2
■5
6.2
I.U
5.Q
1 2.8
11 '' ■'•■'
I .;
10.6 5.2
\U
— £.6
+ 0.9
. - 5-7
+ 2.2
ii
2(, — I 2 . .4
- I.= ■ -12.2 - 4..S
>7
5-2
c. 7
5 5
1.6
27 13.2
'.4 13-9 3.8
H
50
0.6
5.5
I.O
ll 28 13.6
1.2 15. I 2.3
19
5.»
0.4
c.?
•i- 0.3
29 13. s
10 15-7—06
20
■•4
+ 0.2
6.1
- 0.4
3" 13. f'
0.6 13.3 4- I.I
21
- 6.1
0.0
- 6.6
— 1.2
■' 31 1 -13-2
- ".3 -14.2 -r 2.6
22
7.0
— 0.2
7-2
2.2
■N''n-. I 12.4
+ 0.1 12.6 3.6
23
S.I
0. 5
7-5
3.2
2 11.4
0.4 II. I 43
24
9-4
0.5
8.5
4.3
ji 1
1 3 I'l-S
"•7 9-6 4.5
2;
»o.6
I.I
9-3
5.2
4 ; 9-5
o.S
8.4 4.5
20
— II.S
- '3
-10.3
- 5.8
5^-8.5
+ I.O
- 7.4 + 4.3
27
12.7
'•5
11.5
6.0
6 7.7
I.O
6.7 3-0 ■
25
'3-4
1.6
12. S
5.7
7 7.1 :
I.O
6.3
3-5
29
'3-7
'•5
«3.9
4.8
8 6.6
1 .0
6.2
3.1 \
30
13.?
1.4
M.9
3.4
9 6.4
I.O
6.2
2.6
Oct.
I
-13 5
— 1.2
-15.2
- '.7
10 ' - 6.3
+ 0.9
- 6.5
+ 2.1
2
130
0.9
'4.7
-o.t
II 6.3
0.7
6.9
'•5
3
12.2
0.5
■ 13.6
4. ,.4
12 6.5
0.6 7.3
+ 0.8
4
I I . 5
— 0,2
12.2
2.6!
13 6.8
0-4 ; 7.7
— 0.:
S
ro.6
-t- 0.1
10.?
3-3!
14 7.r
+ O.I 1 7.8
I.O
6
- 9-9
-^ 0.4
- 9.4
+ 3.g
»5 , - 7.4
— 0.1 , _ 7,8
- 1.3
7
9-2
0.6
8.4
4-1
16
7.7
■-'•4 7.7
2.7
8
8.7
0.3
7-7
4.2
'7
8.1
(
0.6 7.6
3.4!
9
8.2
I.O
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0.16
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32
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Ncri'.. — Each coluinii
error ever exceeding o.' o;
is compiuecl for the miilclle of the inoiuh, litit in
If much j,'rcater accuracy ihan this is re(|uired
ly 111.' iiseil for the entire iiionlh witliout an
a horizontal interpolation inust lie used.
h
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2
3
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6
7
8
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20
21
22
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24
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27
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