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Full text of "Papers relating to the transit of Venus in 1874, prepared under the direction of the Commission authorized by Congress and published by the authority of the Hon. secretary of the navy. Pt. I-[II]"

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II \v I \i; IF TIM-: MOON; 






I'll I-: I I; APPLICATION 





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4IU6-1958 









INVESTIGATION 



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CORRECTIONS TO IIANSEN'S TABLES OF THE MOON; 



TAltl.F.S FOR THK1R A 1'1'MC VI'ION. 



SIMON NE\VCOM1?, 



IV rt. HAW. 



^ 

FORMING PART III OF PAPERS PUBLISHED BY THE COMMISSION ON THE TRANSIT OF VENUS. 



WASHINGTON: 

GOVERNMENT PRINTING OPPICK. 
1876. 



TABLE OF CONTENTS. 



I'M.. 

|M> > MOTE 5 

# I. INVESTIGATION OF ERRORS OF LONGITUDE. 

I v. I .11 S 

V.uiulion g 

Mean error of uliul.u n^hl ascension al different limes of day q 

Value of solar parallax employed <> 

LiM of com-, nous 1,1 Argument Fondamcnl.il" . 10 

General ideas which foim the basis of this invcMigation 11 

Differential coefficients 12 

Mean apparent error of 1 1 arum's Table* in right ascension 12 

Su.Mcn apparent alteration in mean motion of the moon 13 

<'..ru-. n. in* |.>r limb and observatory to tender observation* siiirtlr comparable 13 

Mean outstanding tabular error of the moon in longitude . . 13 

'rctkms of short period actually applied 14 

K<juaiion connecting errors of moon's tabular right ascension with emu* <>t lunar element*. ....... id 

- of errors of moon's i <>u<-. i-.l n^hl jurcnviin given liy observations al Greenwich and Washington. ... 17 

s -tnal equation* d>r ilnrrmining M, 4, and t by least squares :, 

V .tlucs of outstanding errors of lunar elements for t-.K M \ t ." 20 

Apparent periodic character of the corrections to the ecccnlm ii\ .md perigee 20 

Formula! for the new inequality of longitude 24 

Discussion of Greenwich observations of the moon from 1847 to iSjg ... 94 

Sums of residual errors 26 

ctions to pcccnlriciiv. longitude of perigee, and mojn' longitude 39 

g a. INVESTIGATION OF POLAR DISTAM I AM) LATITUDE. 

Corrections to declination depending on errors of longitude 32 

Constant corrections to reduce declinations to same fundamental standard 32 

Sums of errors of moon's corrected declination, giren by observations at Greenwich and Washington 34 

Correction to inclination of orbit and longitude of node 36 

3. AUXILIARY TABLES FOR FACILITATING THK COMPUTATION OF THE CORRECTIONS To 

IIANSEN'S "TAHI.KS DK LA LI'NE". 

Summary of corrections to mean and true longitude of the moon from Hansen's Tables 37 

K\|>lanalion of tables for applying these corrections 37 

Example of the use of the tables 40 

lions to the Ephemcris derived fiom llansen's Tables of the Moon, for Greenwich mean noon of each day, 

from 1874. September I, to 1875. January 31 41 

Tables I. II. Ill, the arguments g, I*. A. B, u 4j 

Tables IV, V, VI, secular and empirical terms 46 

Table VII. terms of mean longitude 47 

Talilc VIII. u-rmsof true longitude 4$ 

Tables IX. X, factors for reduction to longitude in orbit ; and for correction of latitude and reduction to ecliptic 

longitude Jo 

Table XI. factors for convening small changes of longitude and latitude into changes of right ascension and decli. 

nation i 



INTRODUCTORY NOTE. 



When (lie problem <!' utilizing tlie observations of occultatiuni* nt tin- sevenil Transit 
of Venn* stations, so as to determine the longitudes of those stations with all attainable 
aeeur.iey, was presented tu tin: Commission on tin- Transit of Venus, it was found neces- 
sary t> make i cairl'ul .l.-t.-i ininatiun ol tlic errors of tlie lunar e|iliemeris l>efon- an 
i-ntin-lv satisfactory solution <( tin- |>roli|<-m nml.l In- att-in|iti-i|. Tlie Si-en-tary of tin- 
( '..niinissioii was therefor*- eharijed with this work, most of the eom|iiitations on whirli 
have leen made under his ilin-elion l>y Mr. 1>. I*. Tohl, c-ompulcr for tin; Commisrtion. 

WASIIINOTOM, May 25, 1876. 



CORRECTIONS TO BE APPLIED TO HANSEN'S TABLES 

OF THE MOON. 



INVI--I H. \ I I. -\ OF ERRORS OF I.ONC.ITUDK. 

One of the must important operations in connection with the nhserval inns nt I lie 
Iraii-il nl Venus is (In- accurate determination of the longitude! of the stations. Many 
nt' these st;iliniis arc so fur reim.w-d from telegraphic communication that tin- longitude! 
mn-t .l.-|i.-n.l mainly on tin- moon. Determinations nl longitude from moon culmina- 
tiuns arc lound ly ev|erience to lie subject In constant errors which it is difficult l 
ilctcriniiic and allow for. It was therefore a part nl' the policy of the American < 'oni- 
ini-sion to ilepenil on occnltatinns rnllicr than ii|ni UMNIII culminations for the determi- 
nation nt lnnt;itiiiles. The reason for this course is, that I lie disappearance of a >tar 
liehiml the limli of the moon is a sudden phenomenon, the time ol' which can always In- 
fixed within a fraction of a second. If the cphemeris of the moon and star were cor- 
rect, and the disk of the former a |>erfect circle, the longitude could lie determined 
from the occultation with the same decree of accuracy that the phenomenon could he 
oliM-rved. The question arises, how far these sources of error can be dimininhed. The 
inequalities of, I IIP lunar surface form a source of error which it is im|mssilile to avoid, 
luit \\hich is comparatively innocuous when many observations are made, since lin- 
en-ore will lie purely accidental, and will then-fore lie eliminated from the mean nl a 
great number of observations. 

The position of (he star can be determined by meridian observations with almost 
any required degree of accuracy. We have. then, only to see how far the ern>rs of the 
lunar ephemeris can be diminished: and to reduce these errors to a minimum is the 
object ol the present paper. 

Han-en's tables are taken for this pur|M>se. because there is reason to believe that 
the perturbations on which these tables are founded are, in the main, extremely 
accurate; more accurate and complete, in fact, than any others which have been 
tabulated. Still, before they can be used lor the purpose in question, a number o! 
important corrections are required, which we may divide into twoclas-e-. corrections 
to the theory, and to the element*. 

It is well known that llansen incn-a.-ed all I lie perturbations of his table.- by the 
r.in-tant liictor O.OOOI 544. on account of a siip|Ni>ed \\ant of coincidence bet u ceil the 



8 

Center of figure and the center of gravity of the moon. I have shown that Hanscn 
fails to sustain this position, and that there is no good reason to suppose that the moon 
differs from any other of the heavenly bodies in this respect.* Our first course would 
therefore be to diminish all of Hanson's inequalities by this factor, were it not that there 
are reasons why each of the two greatest perturbations of the moon's motion, the eve'c- 
tionand the variation, should be found larger from observation than he found them from 
theory. 

Evection. The evection has the eccentricity as a factor; the value of the other 
factor being nearly 0.4. If, then, the adopted eccentricity of the moon be erroneous, 
the computed evection will be erroneous by four-tenths the amount of the error. Now, 
by reference to Hanson's "Darlegung dcr iheoretischcn Berechnuttg der in den Mondtu- 
feln angnrandten Storungcn'^ (page 173), it will be seen that the eccentricity adopted 
throughout in the computation of the perturbations of the moon is less by 0.0000073 
than the value lie finally found from observation, and adopted in the tables. Had he 
used the latter value, the theoretical evection would have been greater by the fraction 

-^ =0.000133. The factor actually used being 0.00015 44, ^' e evection, thus in- 
549 O " 

creased, is too large by only 0.00002 1 of its entire amount, or o".O9. Consequently 
the tabular coefficient of evection should be diminished by this amount. Precisely the 
same result follows, if we adopt Hausen's view of a separation of the centers of figure 
and gravity of the moon; and Ilansen himself is led to it on page 175 of the work cited, 
only instead of o".09, he says, "kein voiles Zehntheil einer Secunde.' 1 

Variation. That the coefficient of variation resulting from meridian observations 
will be greater than the actual coefficient may be anticipated from (he following con- 
siderations. The inequality in question attains its maxima and minima in the moon's 
octants. In the first octant, we have a maximum. The elongation of the moon from 
the sun is then about 3 h ; and the observed position of the moon is mainly dependent on 
observations of the first limb made in the day time, when the apparent semi-diameter of 
the moon will be diminished by the brilliancy of the surrounding sky. No account of 
this diminution of the apparent semi-diameter being taken in the reductions, the semi- 
diameter actually applied is too large, and the observed right ascension of the moon is 
also too large. 

When the moon reaches the third octant, the value of the variation attains its min- 
imum. The moon then transits at 9*, and the meridian observation is made on the first 
limb, while the apparent semi-diameter is increased by the irradiation consequent upon 
the contrast between the moon and the sky. The result will be that the observed right 
ascension will be too small. 

The same causes will make the observed right ascension too great in the fifth 
octant, and too small in the seventh. These positive and negative errors of observed 
right ascension correspond to the times of maximum and minimum effects of variation 
in increasing the longitude of the moon. Therefore, the observed variation will appa- 

Proceedings of the American Association for tlio Advancement of Science, 1868. Silliinan'a American Journal 
of Science, November, 1868. 

t Abliaiidlmigen dor mrttli.-nitinch-pliyniclion Cliume der Koniglich SUohumchen Genollschaft der \Vissenncuaften 
Hand vi. 



9 

rently l>c larger than tlir iictual vaiiai \\liali-\rr llii> may I"-. This seems a much 

more natural ami |iriilial>l>- eaii-e lor I In- apparent excess of tin- oliserved over the theoreti- 
cal pertuiliation> llian that assigned liy llanscii. Hansen's factor onl\ increases tin- coelli- 
cient in question liy o ..;;: Imt il .iii- prolialde tliat tin- variation derived I'ruiii oliser- 
\ation- alone \\onlil In- Ml laii/i-r than ll.m-i n'- increased \ariation. In tact, in 1X67, 1 
toiniil. li\- comparin:: tin- error-, ol" tin- lunar ephemeri> when the moon culminated ut 
dilli-rent tinif> nl' tin- ilay, lliat tin- ell'eet of tlic yreater irradiation at iii^lil \\a> \n\ 
stioiiijly marked, huriiiir tin- limr JGUt 1862-65 the mean errors ot the tallies ill 
rijjlil act IIMOII at ilillt-rfiit tunes ol' day u.-n- a> follows:* 

. 

Itflort- Min^rt O. 154 

AHt-r liri^'lit tlayliirlit in tin- cvfiiinij. . 0.093 

I'., ion- liriylit da\liirlit in tin- morning. . . . -{-0.09! 
AlW snnrisf . ... . 4-0. 15.; 

In tin- tlilli-rrncf IM tu. m tin- results liir cai-li limli, (In- fllt-ct ol' incri-ascil irratlia- 
titin M'fins to IP.- . .'.06. 

'I'lir only i. -111.1111111^ I. -mi \\ liich i- laiu 1 -- i-noiiL'li Iti In- materially aiTccli-d li\ tlic 
UK T.MX.- in (|iu xii.Hi is tlic annual ci|iiatioii, ot \\liicli tin- incrca>r is D".IO. 

.inrc at tlic errors of I lan>i-n'> (allies. yiM-ii li\ mcritliaii oli>ei valions, will ti|i>w 

that th.-i ii.nv alitmt the til I lirsl quarter, ami, indeed, dnrin<r tlie first lialf nf the 

lunation, an- in the mean less \>\ l>et\\een ; and 4" than during tlie sc<-ontl half. 
Hence, either the >emi -diameter, or (lie parallactic ei|iiation, or liuth, arc ton lari;e. The 
parallaclic ci|iialion used liy Haiisen corrcs|M)iuls In a value S". 916 for the solar paral- 
lax, which \alnc is too lame liy (irolialdy nut much less than o".lO. The result 
which I dedneed in iS(>7 from all the really \alualde data extant was S .SjS ; ami the 
ilelerminations which have since lieen made, u hen revised witli the lies! tlata, seem tu 
iiitlicate a diminution of this \alne rather than an increase. These indication-, are, hnw- 
a little loo intlelinite to predicate aiiylliiiiir II|NHI. I shall then-tore con- 
tinue to W (S, which will diminish llansen's value l.\ o".o68. The corrcs|ximliiiij 
diminution in the princi|ud parallaclic term will lie <>".(>f>, \\ liile there will he two other 
term- to reeeive a smaller diminution. 



This correclion will slill leave a ilillerenee ol' admit 2" lie! \\een the n>ull> from 

the fust and set I limli-, w liicli will lie accounted for liy an error of i" in the adopted 

semi-diameter. This correction tu the semi-diameter is // ]>n<ni quite prolialde. as the 
improved meridian instruments of the present time i:ive a semi-diameter of the sun i' 
leas tlian tlie ulder ones from which the diameters adopted in our ephemerides were 
derived. It is to lie expected ( hat I he semi-diameter of t lie moon will exhiliit a sim- 
ilar apparent diminution. 

I'icima note in llans.-n'- /'.// 1< -///'^ (page 439), il will lie seen that tine of I lie term> 
in the true lon-itude has crept into the (aides with a wrunir siyn. Asemploved in the talde-. 
and i\en mi pa^'e i^ufthe in! rutliict ion. it is, +o"-335 Kin ( 2 4 gf -|- 20) 4&''). 

As revised ill the l>,lll, L'HHZ. it is . O".28S8ill 

Therefore the taldes neet) the correction ... o".62 sin 

l,n.-t, _;.,!,.. ..I I!,. I leaner or II.. >...! 

IB 



10 

The following is a list of the corrections \vc have so far deduced to Hansen's (allies. 
They should in strictness be applied to the mean longitude, or "Argument fwtliamfntai", 
but they may without serious error be applied to the true longitude. 

Put 

D, the argument of parallactic inequality, or mean elongation of the moon from 

the sun ; 

, the moon's mean anomaly ; 

g', the sun's mean anomaly ; 

co, the distance of the moon's perigee from the ascending node; 

co', the distance of the sun's perigee from the same node. 

We then have 

D = g g' + & oaf, 

and the corrections in question are, 

// 
+ 0.96 sin D ^ 

-f- O.O7 Sin (/? g ) > rarallaetic term* 

- 0.13 sin 

-4- O.OQ 8111 ,' 

^0.33 sill 2 D Variation, 

O. I O si II ( 2 D g) Ktntiim. 

O.62 sill ( g 2 4 s' -\- 2 Csl 4 ft)') Accidental eirnr. 

The fourth and fifth terms of this expression have the effect to remove the increase 
which Hanseii applied to his inequalities on account of the position of the center of 
gravity of the moon, while the sixth is the result of the slight error of the eccentricity 
which he employed in computing the coefficient of evection. 

In comparing with meridian observations which have been reduced without any 
correction to the apparent, semi-diameter depending on the time of day, the correction 
of variation may also be omitted, since a yet larger apparent correction, having the oppo' 
site algebraic sign, will result from the apparent variations of that semi-dinmeler, as 
already explained. 

As regards the possible corrections to the elements of Hansen's tables, it, is to be 
remarked that thai investigator did not avail himself of the elements of the lunar orbit 
deduced by Airy from the (Greenwich observations between 1750 and 1830, but obtained 
his final values of the elements by a comparison of his own. Of the nature and extent 
of the observations tlius employed, we have no details; but it is not likely that more 
than a very small fraction of the entire mass of observations was used, and it can there- 
fore hardly be expected that the elements were determined with (lie hist degree of 
accuracy. Any error in the motion of the perigee or node will constantly increase with 
t lie time. If, in addition to this, we relied, that the meridian observations of the last 
twenty \ ears are tin- more accurate than (hose Hanseii had at his disposal, it, will not 
seem at all surprisinj,' to find quite sensible errors in the present longitudes of the lunar 
periirer and node as derived by Hanseii. Our ne\< step will therefore be to determine 



II 

\v li lion- t.i ||.uiNen'> element* art- indicated liy tin- recent olisei \.i! HUI* of the 

union 111. nli- ;it ( i reen\\ irli ami Washington since 1 863, .1 peiind dmini: which Imth 
.,| oli-ervatioiis an- i aii-fully compared \\itli Han-en'- laldes. 

Tin- irem-ral ideas mi which tin- pies. -lit inxexii-aii I' these corrections i* liMed 

lli.' error* of the 's laliular longitude an- of l\\u class.-... It progressive 

correction, \\liicli apparently increases uniformly w ith the time; and emus of whorl 
period, tin- principal ones nl' which -jo llinniu'li their period durini: "lie n-vnlnlion nl' tin- 
in. i. >n in- lt-. In ilcliTiniiiinx' III' 1 'Tfiii-N nl tin- lir.-l fliws frnni oliMTvalinii. llinsc nl' 
the MTiuiil rla-- ma\ ! i.-^anl.'.l a> acciilciilal i-rrnr>, llif Hi-cl nl which \\ill \te 'lim- 
in. iic . I in. 111 (lie mi-ail u| .1 laii.'<- iiiiinlirr nl' uli-rn al loiis. Since, in a series nl nUserva- 
tinti> i-\iriiiliiiir tlimnuli a iniiiilier nl' \eais. (lie maxima ami minima nl each term of 
imrt |terin<l \\ill fall iinliscriminateU mtn all |iarts nf all the other |erinils, each periodic 
eiirreetiiin may lie uetermined as if the ell'ecls ofllir itliei> \\en- jiuiely acciilental 
MI.''- At (he Mime time, as the elimination of each periodic error Irom the maxima 
.iiul minima nl' all the nlheo cannot Ke complete in any finite time, it is desiralile that 
each |H-rioilic cnrrectinn nl sensilile maumhnle hicli v\ e can determine lielorehand shall 
lie a|i|died to the residuals liefore the latlei are Used to determine the cnrrectinlls to the 
element-. 

The corrections of the elements of longitude have lieen made to de|iend |n incipall y 
UJMIII tin- oli--r\ed riuht a.-censinns, instead of reducing ihe oliser\ed errors of i i^'hl 
. nsimi and polar distance to errors of longitude and latitude. The lea-mi lor this 
coiir-i IN. iiul the a|iparent emus of polar distance, after correcliiiu' them approximately 
fnr errors of the elements easily determined, u ill aiise principally from error* of ..! 
\alinn, and not In MM ermrs nf the tallies. In fad. the oliset \ations of the IIIIHIII'S declin.i 
lion an- sometimes allected with accidental errors of a magnitude \\hieh it is ditlicult to 
account for, especially in the case of Wa>hini;lon. (iraiitiny that Ihe moon move* in a 
plain- the |M>sition of which can lie very accurately determined, we liave aOerward only 
to determine the moon's |M>sitioii in that plane, and this can lie done from an olisened 
right aM-ensioii almost as well ax if we had a direrlly olisened loogittlda The lonyi- 
tude thus determined will In: less likely to lie allected \\ilK systematic errors thai) if we 
siip|Hise the position entirely unknown, and change (hi- errors of rinht ascensi(n and 
declination In errors of loimitude and latitude, without regard to the |MissiMe constant 
errors nf the measured ilerliiiations. 

Formula- for expres>int: (he longitude and latitude nf the moon in terms of the 
lunar elements are liiven liy llansen in a piistlinnioiis memoir.* The following; ti-rm- 
are siitlicient for our present purpi- 

I'ut 

/, tin- mnmi's Imiiritiide ill mliit ; 
6, the lnni;itnile of the ascendini; node; 
*, the inclination of the nrliit to the ecliptic: 
\ the moon's ri^'lil ascension and declination: 
tin- oliliiplity ot' the ecliptic. 

I ..l-i .1..- HarmU-lluiii; .1. r _-t ..|. :, \ ,.I>II-I K IIIIK nn.l AbwrirtiunK do M : UMtiM ttt lAff ! 4cT Bbo 

and .1, r KnoUnUoRr. Abli.llui^"ii drr Kopiglinh BlihriMhiB OIUebft dcr WiweuKlicArD, lid. i, Vo, Till. 



12 



We then have, approximately, 

a I 2.5sin 2/ i.i sin(2/ (9) -f i.i sin 
sin S sin oo sin I + cos ca sin i sin (/ 0) 

0.40 sin / -f 0.08 sin (7 <9) 
The differential co-efficients derived from these expressions are, 

i 0.037 cos (2 / 0) 0.087 cos 2 / 
dl 

''"' o.oi8cos0+ooiScos(2/ 9) 
^- 0.2 1 sin 9 0.2 1 sin (2 / 9) 



cos S 



cos 



0.40 cos / -f- 0.08 cos (/ 0) 

= (0.40 -f 0.08 cos 0) cos I + 0.08 sin 9 sin , 

= 0.08 1 cos (I 9} 



dS 



cos S - 0.92 sin (/ <9) 
r/t 

From the first three ibrmula 1 , it will be seen, that the mean error in right ascension 
is very nearly the same as the mean error in longitude ; (lie periodic corrections being 
supposed to lie eliminated from this mean. 

The investigation of the corrections from observations is now made as follows: 
All the apparent errors of the tables derived from the meridian observations at Green- 
wich and Washington since I S62 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 errors of Hanson's tables in ft. A. 



Greenwich. 




Washington. 


Mean. 


"Year. 


L 


II. 


Diff. 


I. 


II. 


Dill. 


I. 


II. 


Mean. 


1862 


' 


II 











" 


- 3-6 


0.6 


2.1 


1863 














2.3 


+ 0.5 


-- o.o 


1864 














I.O 


+ 1.3 


+ 0.4 


1865 


0.2 


+ 3-o 


3-2 


+ 0.3 


+ 3-9 


3-6 


o.o 


+ 3-4 


+ 1.7 


1866 


+ 1.2 


+ 3-6 


2.4 


+ 0.9 


+ 4-5 


3.6 


+ I.O 


+ 4-0 


+ 2.5 


1867 


+ 2.4 


+ 5-7 


3-3 


+ 2.4 


+ 5.8 


;-4 


+ 2.4 


+ 5-8 


+ 4.1 


1868 


+ 2.6 


+ 6.0 


3-4 


+ 2.4 


+ 6.6 


4.2 


+ 2.5 


+ 6.3 


+ 4-4 


1869 


+ 3-3 


+ 5-6 


2.3 


+ ? 


+ 7-4 


4.0 


+ 3.4 


+ 6.5 


+ 4-9 


1870 


+ 3-4 


+ 6.6 


3-2 


+ 4-6 


+ 72 


2.6 


+ 4-0 


+ 6.9 


+ 5-4 


1871 


+ 5-4 


+ 8.2 


2.8 


+ 5-1 


+ 7-8 


2-7 


+ 5.2 


+ 8.0 


+ 6.6 


1872 


+ 6.0 


+ 8.7 


2.7 


+ 6.2 


1 '!.<> 


3-4 


+ 6.1 


+ 9.2 


+ 7-6 


1873 


+ 6.9 


+ 9-4 


2-5 


+ 6.9 


+ 10.2 


3-3 


+ 6.9 


+ 1O. 2 


+ 8.6 


1874 


+ 8.1 


+ 11. 4 


3-3 


+ 7.1 


+ 10.8 


3-7 


+ 7-6 


+ 11. 1 


+ 9-4 



The last column exhibits the apparent tabular errors in mean right ascension, and 



L8 



the if loi f in ii it -.in longitude. ;i> .If IIM d f.u-h year I mm .ill tl Itserxatmns. The siulilfii 

.I|>|MI. nl .ill. i.iin. M nl in .iily inn- M-fiiinl per .ininiiii in the mean motion oi the moon, 
exhiliiled in (Ills Ciilliinii, seems tu nif mif of tin- must f\l r.ionliii.ii y of astronomical 

plici if n.i : luit, ;IN | II.IM- diM-iis*.,-,! il in M-MI.I! p.i|>fi- .luring the l;i>t I'm- yearn, I 

shall <ln mi iiinif here than rail attfiitimi to its continuance, .in. I to the ini|Hissiliility of 

lepr.-M-ntiii;,' ii liy any MIL ill number o( ) liuilu- (elms w itliont introducing diaoonkuM 

int. i tin- lonu'ilude during prc\ ion- 
It will In- seen that then- an- iliscordaiief.N lietweeii tin- re>nlts of tin- t\\o uli>fr\a- 

loin >, > liiiif- aiiiiinniiii^ tu IIHIIV than a >i-rnii,l. In ilfifiiiiiiiin^ tin- roncftiuns <if 

short |ifiii>il. it i> ilf-iralili: tu rciluri- tin- s\ >tnn.ilif fimrs c \lf mlinir tlmillgli rach 
M-at to a minimum; lln i|iif-lion whctlifi such rrmrs arc* in the theory or the ulisfrva- 
tiuiis linn;,' imlillf rent. It i> also ilcsiralilf that in taking the mean of the results of the 
two oliM-i vatorifs, they .-liouM lie made eomiKiralili 1 with each other liy correct iny either 
uf them I'ur the s\ste in;ni<- ilillfrencf. These currrctions, of eom>f, only adinil ul 
a|'|iro\imate ih- termination, .mil they II.IM- In-en applied eat h year tu that olisi-rsatur)' or 
that limli ot'the moon in whtcb, judging lioni the di-vuitions Iruin iinitiirin |iru^ressiun, il 
\\as jiid^'fd most likely that the disrurdaiiee existed. The foOowing are the eorreetions 
a-tuall\ ;i||ilied to the > \.-r.il ela.-si-s ul' l.il.nl.ir f 





(II I Mil U II. 


U 


\ , 








1. 


II 


1. 


II. 


a. 








1862-63 


+ ' ' 


-f 0.06 


u 


- 


1864 


o 


o 


u 


o 


1865-68 


o 





u 


- 0.04 


- 




+ 0.06 





- 0.04 


1870 


+ 0.06 


o 


o 


- 0.04 


1871 





o 





o 


8 7 a 





o 


o 


0.04 


1873-74 





o 


o 






Haying applied these corrections throughout their several years, the ( iieenu ieh 
and Washington obMrtatioiU were eonsidered strictly COmpMable; and when the moon 
\\.i- i.l.xi r\fd ;it I. nili uliM r\;ilorii - mi the same, day, the mean ul' the corrected talmlar 
errors \\.i- taken. Tin- mean uiitMandini: talmlar error for each year now In-conic* aa 
follow - : 



Year. 


Yew. 


1862 


2.1 




1863 

1X64 
I86 5 


-0.9 
+ 0.4 

-f 1.4. 





-f-2.2 
-f 

+ 4-' 



f, u 



A 



1870 



-f 5-' 
+ 5-6 
+ 6.6 



Year. 



+ 8.6 
+ 9-7 



These i|ii;intitics. with the si^'ii ehaie.'ed, simuM lie considered as currectiuns tu the 

fundamental argument, and we have tu determine the eorres| diiiL' collection to the 

ii^lil aceiisions which are to ! applied tu the indi\idiial talmlar rrn-r-. Tu 
them to coiifdioiis of true lunirilude, they an- to lie multiplied liy the factor 

I -f- 2 I' CO* g =. I -f O. I 1 CO g 



14 



Tfog corresponding factor for Correction of riglit ascension is, with sufficient approx- 

imation, 

da (i -f-o. i i cos g 0.04 cos (2 / 6) 0.09 cos 2 /) 6 A 

III tlii.s formula, eSA represents tlit- correction to the mean longitude, while we may 
suppose / to represent indifferently the mean or the true longitude; and, during a period 
of several months at a time, we may represent the longitude as a function of g. The 
value of da has been reduced to a table of double entry as a fund ion of g and of tin- 
time. To express the mean longitude; as a function of g, we have 

I- g + 7T 



where 



2 I =. 2 g -}- 2 7t 

\\j the substitution of these values, the expression for 80. becomes 
8a ( j -j- o. 1 1 cos g + A cos 2 g -f- B sin 2 g) 6\ 



A .04 COS (2 7T 9) .09 COS 2 7T 

B .04 sin (2 TT 0) + .09 sill 2 7t 
The values of /T, d, A, and B for periods of six mouths are as follow 



Year. 


ir 





A 


B 


Year. 


rr 


a 


A 


a 


1862.0 


o 

228 


274 


+ .05 


+ .09 


1869.0 


153 




"39 


.01 


.06 


1862.5 


248 


264 


4- .oq 


+ .09 


1869.5 


'73 


129 


.07 


- .05 


1863.0 


269 


255 


-1- .08 


.04 


1870.0 


'94 


119 


- .08 


.00 


1863.5 


289 


245 


4- .03 


- .08 


1870.5 


214 


110 


- .06 


+ .05 


1864.0 


309 


235 


.02 


- -07 


1871.0 


234 


100 


.01 


4- .09 


1864.5 


330 


226 


- -05 


- -4 


1871.5 


255 


90 


4- .06 


4- .08 


1865.0 


350 


216 


- .06 


.00 


1872.0 


275 


81 


4- .10 


4- .02 


1865.5 


310 


206 


- -05 


4- .03 


1872.5 


295 


7i 


4- .09 


- .06 


1866.0 


31 


>97 


.01 


4- .08 


1873.0 


316 


61 


4- .04 


.11 


1866.5 


51 


187 


4- .02 


4- .05 


1873.5 


336 


52 


- .05 


. II 


1867.0 


71 


177 


4- .05 


4- .03 


1874.0 


356 


42 


.12 


.04 


1867.5 


92 


168 


4- .05 


.00 


1874-5 


7 


32 


.12 


4- .05 


1868.0 


112 


158 


+ .04 


.02 


1875.0 


37 


23 


.04 


4- .12 


1868.5 


133 


148 


+ .03 


- .05 













The coellicient i -f o. i i cos i- -}- // cos 2 g + .Bsin 2- is next tabulated for each 
of these sets of values uf . / and />' for every IO of g, and multiplied by the corre- 
sponding value of <5A. As these tables are superseded by those given at the close of 
this paper, it is not necessary to print them. 

The rorrecticms of short period, which have been actually applied, are 

// 

+ 0.96 sin D 

-o.i;, sin (D + g 1 ) 
-f- 0.09 sin x' 

- 0.62 sin (2 g 4 g' -f 2 u> 4 a/) 



15 



Tin- lii>t llin i- li.iM- IM-I-II ( omliinril lulu a -iii^l'- "in n|' i|inilil<> ai u'limi nl . in \\ liirli 
tlic ari!iiiiirnN an- l> ainl tin- month; (In- latter C(irrcs|MHi(lilin t.i i.' . The li-mis ( |r|.-nil 
tit on this arumnriit .>" -" -mall lli.it they m.i\ In- i .--.n ii.-.l .1- rmislant during nil 
rntirr liinlitll. 

Ill tlii^ ..inn- t.iUi- i> inrlinlril ;i partially ronjrrlural nm rrtioii tiir tilt- variations ol 

tin- n I's si-mi-iliamrtri. Tin- ronvctiou to Haulm's \aln<- lias lu-cn assumed M 

2".o, \\lirn tin- moon is in tin- m-inhliorli I of the sun, >.. that IHT linili is MTV I'ainl. 

and us o" ..\ alter the close nf evenim; t\\ilii.'hl. I let \\een t \vii hours ol ' rlonuat ion 
ami tin- rlox,- ,,|" t u iliiiht. it is a.-Mimed to increase uniformly. Tin- .sum of tin-si- four 
correction-. i> nixeii in the lollowini: talilc : 





i iK-r I.IMII. 


|| 


V e c 




Jj c c 


Q 


Ian. 




Mar. 


Apiil. 


May. 


lunr. 


July. 


Aug. 




Oct 


Nov. 


P.. 


|H 




















' 


V 


M 




ii 




X 
























11 




























12 




























1 1 






















4-17 


+ 17 


+ i 8 




- 9 


+ 1.8 




* 

f 1-3 


-r 1.3 


+ S.I 


+ l.l 


+ l.o 


t i.-i 


+ 1.8 


+ 1.6 

* I J 


111 


4. i c 


- Q 

g 


- 7 


+ 5 


+ 15 


+ 1.8 


+ 1.8 


+ 1.8 


+ 1.8 




H.6 


+ 1-4 




+ 1.4 


>r 1.5 


- 7 


- 6 


+ 1.5 


+ 4 


+ 1.4 


+ 1-5 


+ 1.6 


+ 5 


+ 1-5 


+ 1-4 


+ l.l 


+ 1-3 


-f 1.4 


+ 1.4 


- 6 


- 5 


+ 1.4 


+ 1-3 


+ i 3 


+ l.l 


+ 1-4 


+ 1.4 


+ '.3 


+ I.I 


+ 1.1 


+ 1.8 


+ 3 


+ 1.4 


- 5 


4 




















4 1 o 


4- 1 1 


I > 


t 


1 


+ 0.9 


+ 0.8 


+ 0.8 


+ 0.8 


+ 0.7 


4- 0.7 


+ 0.8 


+ o.S 


+ 0.8 


+ 0.8 


+ 0.9 


+ 0.9 


2 


1 


f 0.6 


+ 0.6 




+ 0.6 




+ 0.6 


t 0.6 




f 0.6 


+ 0.6 






I 

o 












s 








. 








\\ 


II 

' = = 




F.COND I.I MM. 


t: 


Jan. 


Feb. 


Mar. 


A,, i,l 


May. 


June. 


July. 


Aug. 




Oct. 


Nor. 


Dec. 








.. 




M 

























-0.4 


-0.4 


-0.4 


-0.4 


-0.4 


-0.4 


-0.4 


-0.4 


-0.4 


-0.4 


- "-4 


-0.4 





+ 1 


-0.6 


-0.6 


-0.6 


-0.6 


-0.6 


-0.6 


-0.6 


-0.6 


-0.6 


-0.6 


-0.6 


-0.6 


+ 1 


+ s 


-0.8 


-0.7 


-0.8 


-0.8 


-0.8 


-0.8 


-0.7 


-0.7 


-0.8 


-0.8 


-0.8 


-0.8 


+ t 


+ 3 


1.1 


-0.9 


l.o 


l.o 


-0.9 


-0.9 


-0.9 


-0.9 


- l.o 


- l.o 


1.0 


1.1 


+ 3 


+ 4 


i.l 


- i.l 


1.1 


i.l 


i.l 


1.0 


l.o 


i . i 


l.l 


- I.l 


- i.l 


- l.S 


4- 4 


+ 5 


- 1-4 


- i.l 


- l.S 


1.1 


- -4 


' 


- 1.4 


l.S 


- -3 


-1 3 


- 1-4 


-1.4 


- 


+ 6 


- 1.4 


-1.3 


- -3 


- 1.4 


- -5 


- '-5 


- 1.5 


--5 


- 1-4 


- 4 


- 1-4 


-'5 


+ 6 


+ 7 


- -S 


- -3 


-1.6 


-1.6 


-1.7 


- -7 


-1.8 


- 1-7 


-1.6 


- -5 


- -5 


- -S 


+ 7 


+ 8 


- 1.4 


-1.5 


-1.8 


-1.8 


-1.9 


-1.9 


l.o 


-1.9 


- 1-9 


--7 


- 1-5 


- -5 


' 


+ 9 


-1-7 


-1-7 


l.o 


- -9 


-l.o 


1.0 


S.I 


l.l 


l.l 


-1.8 


-1.8 


-1-7 




+ 10 


-1.9 


- 1.9 


- 1.0 


l.o 


S.o 


1.1 


- J.I 


8.8 


-*-3 


l.l 


l.o 


-1.9 




+ n 


-1.0 


-1-9 


i.l 


l.o 


S.o 


8.1 


- 1.1 


-8.3 


-8.4 


-i.l 


l.l 


- 8.0 


+ n 


+ 11 


1.1 


-8.0 


S.I 


l.o 


- l.l 


1.1 


-1-3 


-8.4 


- -4 


-8.4 


-8.3 


8.1 


+ IS 


+ "3 


- 1.1 


- i.l 


1.1 


8.0 


1.1 


1.1 


-8-3 


-a 4 


-8.5 


-8.5 


-8.4 


-8.3 




+ 14 


-..3 


l.l 


S.I 


S.I 


l.l 


-a-3 


-8.4 


-8.5 


-8.6 


-8.5 


-.s 


-1.4 


I. 



16 

By the application of the foregoing corrections to tin: errors of the moon's tabular 
right ascension, these errors may be supposed to be reduced to very small quantities, 
depending on the errors of the lunar elements, with which they are connected by the 
equation 

dot , da. da. . 

Sa ~ 61 + -60 + _ Si, 

til d9 di 

the differential coefficients having the values given on page 12. When we substitute 
these values, the expression for da will contain the terms 

(+.018 SO .037 So) cos (2 1 0} 

.087 da cos 2 / 
+ .018 69 cos 
+ 0.21 Si sin 9 

0.2 1 Si sin (2 / 6) 

If we represent the sum of these terms by P, we shall have 

SI = Sa P 

In the investigation of the corrections to the moon's eccentricity and longitude of 
perigee, the terms of P maybe entirely neglected. This arises from the circumstances 
that the appreciable terms of I or a arising from the errors of these elements have the 
same period with g, the mean anomaly, while P contains no appreciable periodic term 
depending on g. The outstanding portion of Sa probably averages not more than one 
second or two at the utmost, so that the term .037 Sa is quite insignificant. The term 
.018 SO may have a constant value of o".25, more or less;* but the short period of the 
term 2 I 9, and its incommensurability with the period of ' g, permit of this error 
being regarded as fortuitous. The same remark applies to the terms .087 Sa cos 2 I 
and 0. 2 1 di a\\\ (2 I 0). The only remaining terms have the period of 0, which is 
more than eighteen years. The effect of these possible errors is then-lore eliminated 
in the mean correction for each year, which has been already applied to the errors. 

To determine the correction to the eccentricity and longitude of the perigee result- 
ing from each year's observations, the residuals in right ascension, after the application 
of the three corrections already described, have been arranged according lo the values 
of the mean anomaly to which they correspond. The results are shown in the follow- 
ing table, which gives for certain limits of mean anomaly in the first column, firstly, the 
sum of the residuals (tabular minus observed) in right ascension, 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 observatories are counted 
separately, so that when observations were made at both observatories on the same 
date, the sum of the residuals is taken, and the observations count 2 in the column N. 

* It i afterward found that, the vulno of tliift product is only o".o8. 



17 



Sums of errors of moon's corrected r'niht <ucaut<" -///> <// ti rerun > 

hinqton. 





IS6* 




1863 




1864 








Limits of mean 


















anomaly. 


I.U 




H 


\ 




N. 


Urn 


N. 




10 10 


+ 3-9 


4 


* 21.5 


10 


+ ' 


9 


1-4 


7 


10 10 *0 


-1- 3-6 


6 


+ ' 


12 


+ 


7 


* 


4 


o to 30 


0.2 


$ 


+ 14 2 


8 


+ S-8 


S 


- 0.3 


10 


JO 10 <0 


+ 9-3 


8 


+ 23.7 


II 


4-5 


7 


- o.$ 


S 


40 lo 5 
50 lo 60 


+ 0.3 


8 


+ 9-8 


9 


- 1.6 


10 


I.I 


6 


60 to 70 


+ 8.9 


10 


- 4-3 


7 


+ 


5 


- 6.1 


7 


70 to So 


- 3-7 


4 


+ 7-0 


10 


- 7.0 






6 


80 lo 99 


+ 6.7 


7 


- 6-7 




- 11.2 


9 


- 




90 to too 


+ 3-9 


6 


- 3-3 


9 


- 3-4 


6 


- 8.$ 


7 


too to no 


+ 3-9 


it 


- 0.4 


5 


- J.I 


S 


- 0.7 


S 


1 IO to I2O 

1*0 to 130 


- 3- 


8 


- 3-9 


7 


+ O.I 


5 


- 5-5 


6 


130 to 140 














+ II 




140 lo 15^ 
150 to 160 


o.i 


S 


- 18.2 




- 


7 


+ > $ 


4 


160 to 170 


- 8.8 


4 


- 19-7 


6 


+ 2.5 


6 


+ 4.3 


5 


170 to 180 


- $-7 


4 


- 9-9 


7 


- S3 


5 


+ 


6 


180 to 190 


- 17-4 


9 


- 33 


14 


- 8.6 


7 


-f 8.9 





100 10 300 


- 15 5 


7 


- -3 


4 


- 0.6 


4 


+ 13-2 


8 


too lo 210 


- 3-8 


10 


.0 


6 


- 6.4 


9 


+ 7-8 


8 


210 to MO 


- 0.2 


2 


- -9 


9 


- 2.9 


8 


+ 3 


7 


MO 10 930 


- 28.9 


. 


- -5 


10 


+ 3.'. 


7 


+ < i 


S 


130 to 240 


- 7.3 


7 


- 9 


7 


+ 0.8 


7 


+ 10.3 


S 


140 to 250 
150 10 260 


s.o 


4 


+ 7-6 


8 


+ 11.5 


8 


+ 7-3 

j. 16 


7 

12 


too to 27 
270 to 280 


37 


5 


+ 11.3 


9 


+ 25.3 


ti 


r **. * 

+ 7-6 


It 


- 


+ 4-7 


7 


t- 0.8 


S 


rt.i 


- 


+ 9-6 


- 


200 10 300 


- -3 


i 


+ 5-9 


- 


+ 6.6 


4 


+ $-8 


II 


300 to 310 


+ 3-o 


J 


+ 23.5 


9 


+ 7-8 


6 


+ 10. 1 


7 


310 lo 320 


+ 2.3 


2 


+ 22.6 


6 


+ 6.4 


S 


-t- 16.4 


10 


320 lo 330 


- 2.8 


: 


-t- 18.2 


9 


+ i 


7 


-f- 14-5 


7 


330 to MO 


+ 9-5 


6 


+ 1.2 


7 


+ 18.5 


10 


+ l6.T 


u 


340 to 350 


-f- 11.8 


- 


+ 7-2 


- 


+ 4 * 


7 


+ 7-6 


7 


350 to 360 


+ 13.6 


S 


+ 14-4 


8 


+ 16.5 


6 


* S-J 


9 




+ 106.4 


22$ 


-t- 222.0 


287 


+ I8T-I 


236 


+ 205.9 


255 




116.0 




-144-7 




- 71.0 




- 46.6 






- 9.6 




+ 78.J 








+ S9-J 





3M 



18 



Sums of errors of moon's corrected right ascension, Sfc. Continued. 





1866. 


1867. 


1868. 


1869. 


Limits of mean 










anomaly. 




















<Ja 


N. 


Ida 


N. 


<Ja 


N. 


Z(!a 


N. 


o to 10 


- 1-7 


6 


II 

+ 7-4 


5 


4-2 


4 


- 10.7 


4 


10 to 20 


- 2.5 


4 


- 5-0 


2 


+ 3-9 


7 


4-2 


4 


20 to 30 


- 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 


4 


- 9.0 


5 


+ II. O 


6 


50 to 60 


- 0.7 


I 


2.O 


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 too 


+ 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 


I2O 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 


+ 10. I 


9 


+ 1.9 


7 


+ 5-2' 


7 


+ 3-1 


6 


150 to 160 


- 4-1 


6 


- 2.6 


7 


+ 2.1 


9 


+ 20.3 


7 


160 to 170 


+ 3-3 


7 


+ 6.8 


5 


+ 1-3 


8 


+ 3-7 


3 


170 to 180 


O.I 


7 


- 5-0 


8 


+ 0.8 


7 


+ 12.2 


7 


180 to 190 


+ 0.8 


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. I 


5 


210 to 220 


+ 0.3 


6 


- 1-7 


4 


+ 13.0 


8 


+ 12.2 


5 


22O tO 230 


- 5-4 


4 


+ 12.9 


9 


+ 4-8 


4 


+ 12.3 


7 


230 to 240 


+ 4-1 


8 


-f- 8.2 


6 


+ 15-2 


9 


- 1-3 


3 


240 to 250 


- 1.8 


7 


+ 25-4 


9 


+ 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 


+ 11.7 


6 


- 5-0 


2 


- 17-3 


7 


270 to 280 


+ 9-7 


4 


+ 3-3 


4 


+ I.O 


7 


18.8 


5 


280 to 290 


+ n. 6 


12 


+ 7.0 


7 


- 9-i 


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 


2 


- 4-8 


2 


310 to 320 


+ 3-4 


2 


+ 2.3 


5 


- 13.. 8 


8 


- 0.8 


t 


320 to 330 


+ 7-7 


5 


+ 0.2 


5 


10.6 


9 


- 4.2 


2 


330 to 340 


+ 9.1 


5 


+ 3-5 


6 


- II- 7 


6 


- 18.5 


6 


340 to 350 


+ 10.8 


6 


5-4 


7 


- 9-8 


5 


10.6 


4 


350 to 360 


+ 92 


7 


- 7-2 


4 


- 18.3 


6 


2.2 


5 




+ 132-9 


213 


+131.2 


182 


+ 161.8 


229 


+ 178.9 


187 




- 54-0 




- 55-9 




115.6 




-139-2 






+ 78.9 




+ 75-3 




+ 46.2 




+ 39-7 





IS* 



SHMU of error* of tuoon't corrected right ascension. Jp. Concluded. 



Umiuofmean 


1870 




1871 




1871 


1 


87> 




1874 




anomaly. 


Zrf 


N. 


Z4* 


N. 


X<U 


N. 


ZJa 


N. 


Irfa 


N. 



OIO IO 


- 7. 


S 


H 

- 3 


5 




4- 6.5 


6 


- 4-3 


6 


4- 


4 


1010 20 


- 2.2 


S 


4- 1.7 


n 


4- 8.5 


IO 


4- S. 


4 


t" 5-9 


5 


010 JO 


4- $ 


6 


- 0.3 


7 


4- 5-5 


8 


4- S- 


8 


4- 12.5 


6 


jo to 40 


4- 10.7 


8 


4- 6.4 


7 


4- II. 8 


7 


4- 3-4 


3 


4- 5.1 


| 


4010 jo 


+ It. 2 


8 


4- 16.7 


9 


4- 6.0 


4 


4- 6.6 


4 


4- 4-4 


5 


so to to 


- 7- 


S 


4- 9-7 


6 


4- 13.2 


6 


4- 4-1 


7 


4- a.i 


S 


to 10 70 


+ 1.0 


9 


+ 18.9 


8 


4- 10.4 


3 


4- 13-4 


6 


4- I 


4 


7010 to 
to to oo 


- 2.6 
4- 12. 


5 

it 


4- 10.2 
4-117 


7 


4- 12.4 


8 


4- 13-5 


3 


4- 6.6 


6 


T^ 
go lo too 


+ IO.I 


8 


* / 

4- 12.5 


3 


4- 9-8 


4 


4- 5-1 


a 


4- 5-9 


7 


too lo no 


4- 10.8 


4 


4- 19-7 


8 


4- 13-0 


6 


4- 1.5 


3 


4- 10.9 


6 


no to 120 


4- 5-8 


6 


4- 8.1 


4 


4- 18.7 


6 


4- .3 


a 


4- 4.6 


4 


1 JO IO I JO 


4- IO.I 


7 


4- 9-7 


5 


4- 18.3 


7 


4- .1 


5 


- 4.7 


6 


13010 140 


4- IO.I 


S 


4- 15.4 


5 


4- 0.2 


a 


4- -3 


3 


4- l.B 


3 


140 10 150 


+ 18.2 


8 


4- 2.1 


3 


4- 2.9 


3 


4- .4 


5 


- 0.8 


7 


150 to 160 


4- 4-4 


3 


4- 3-0 


7 


4- 2.1 


8 


- -9 


3 


4- 1-3 


S 


ito to 170 


+ 8.8 


S 


4- 8.7 


4 


4- 6.6 


5 


.4 


4 


- IO.I 


9 


17010 ito 


+ 6.9 


3 


4- 6.2 


6 


1.2 


3 


- -7 


3 


1.0 


6 


180 10 190 


+ I 8 


i 


4- 3-9 


4 


4- 1.9 


4 


.2 


4 


+ $.0 


6 


190 lo too 


4- 7-5 


4 


4- 3-5 


3 


1.2 


5 


- .6 


6 


1.0 


2 


900 10 210 


4- 2.1 


5 


+ t.o 


3 


2.2 


6 


- -9 


a 


4- 3-7 





aio lo no 


- *-3 


2 


- 2.6 


a 


4- i. a 


3 


- .6 


3 


- 5-0 


5 


MO 10 330 


- *$ 


3 


- 9-3 


7 


- 7- a 


5 


O.I 


4 


16.0 


7 


230 to 940 


- 0.4 


S 


- 3- 


6 


- 4-8 


5 


- 35 


1 


- 3-5 


4 


140 to 250 


- 9-7 


5 


- 9- 


8 


- 6.5 


3 


- 7-5 


5 


- 5-l 


8 


250 to 260 


la.i 


6 


- 5 


5 


- 9 


4 


- 7-1 


4 


- 33.0 


5 


ato to 770 


- -3 


2 


- 4-6 


5 


- 13.8 


8 


- 8.6 


3 


- 22.6 


4 


27010 ato 


- 12.9 


8 


- 7-1 


7 


8.4 


5 


- 4-3 


4 


- 15.6 


4 


ato to 290 


- S-6 


3 


- *-7 


6 


- 16.7 


9 


- 10.8 


6 


- 9.1 


3 


29010 300 


- $-S 


4 


4- 4-0 


4 


- 10.3 


8 


- 9.8 


4 


- 13-4 


8 


300 to 310 


- 4.0 


4 


- 9-5 


6 


- 9-5 


5 


- 1.8 


1 


- 9-' 


9 


310 to 320 


- 8.7 


3 


- 6.6 


S 


- 5-6 


4 


- 3 


4 


- 5-3 


6 


320 to 330 


- 3-5 


6 


- 4-9 


7 


- 8.5 


5 


- ".3 


7 


- 1.4 


7 


33010340 


- 9-7 


4 


- 2.8 


7 


- 8.5 


5 


- o.j 


8 


- 4-3 


3 


340 to 350 


- 3-6 


3 


- 1-7 


4 


- 5 


5 


- 9.2 


6 


4- a. a 


n 


350 to 360 


- 8.7 


5 


4- 6.3 


4 


4- O.I 


6 


- 4.0 


5 


4- 2.5 


6 




+ 136.6 


185 


4-179-5 


*>3 


4-160.4 


95 


4- 96.9 


55 


4- 95.2 


200 




-120.6 




- 71.8 




-II8.6 




-113 I 




-171.0 






+ 16.0 




4-106.7 




+ 41. 




- 16.2 




- 75-8 





Neglecting all terms multiplied liy the eccentricity in tin- c.-mYinits. nidi : 
mil gives an equation of the form 

Jl -f- 2 sin g dp 2 cos'^ e dir =. r 



20 

or, putting 

It zz 2 Jde 2 Se 

k 2 .Je 6V 2 e (Szr 
the equation will be 

41 -\- h sin g + A cos g r, 

-Je and dn being the errors of the tabular eccentricity and longitude of the perigee, 
while Se and STT represent the corresponding corrections. 

The equations are now solved as if all the residuals within each pair of 20 limits 
corresponded to the mean of the limit, that is, as if all between o and 20 corre- 
sponded to g = 10 ; those between g 20 and g 40 to g 30 ; and so on. If, 
then, we put 

gi 10 ; g 2 = 30, etc. ; 

r t , the sum of all the residuals in any one year corresponding to g g t ; 

n it the corresponding number of observations; 

s { = sin gt ; 

Ci = cos gt : 
the normal equations for determining dl, h, and k, by least squares, will be : 



{ ^) Jl + (2 n t s?) h -f (2 n t s t c t )k = 2 s { r { 
(2 n t Ci ) 41 + (2 n, Si d) h + (2 n { c, 2 ) k - 2 e { r t 

The formation and solution of these equations for each year give the following 
values of the outstanding errors of the lunar elements for each year: 

// /' 

1862, /* -f 0.04 = +1.23 

1863, -0.64 +1-78 

1864, 1.07 + 1.09 

1865, -1.03 -0.15 

1866, -0.47 +0.10 

1867, 0.93 0.36 

1868, +0.34 - 1.46 

1869, + 1.67, 1.56 

1870, -j- 1.48 - 1.14 

1871, +1.65 0.36 

1872, + 2 -'5 O.I 2 

1873, + I-9 1 +0.16 

1874, -(- 1.92 +0.60 

The periodic character of thcsi; residuals is very remarkable, indicating, as it does, 
either a hitherto unknown inequality of the moon's mean longitude, having nearly the 
same period with the orbital revolution; or one of < he eccentricity and longitude of 
perigee, having a period of between fifteen and twenty years. To investigate this in- 
equality, we shall assume that each value of h is of the form 

k a sin (JLI + nt) 
and each value of k of the form 

k+ a'cos(yu' +M'/), 



L'l 

//. ft, . n . n, ;ui<l // IM-UIU' unknown quantitie> i.. ) determined, and / the time 

in \.-;u> tiiiin an\ assumed i-|iu<-ii. \\ < shall lake for the eporli tin- inidillr of I he 
period through whieh the obsenations extend: that is, 1868.5. IT. then, \M- represent the 
thirteen values of // and k in chronologic-ill order liy A_,, A_,, ..... A,, *_*_* ..... 
tin- equations of condition lor It and k respectively may In- put into tin- form 

A, zz A a sin // - / n a cos // sin n 
k t rz k + a' cos ft r..s / n a' sin // sin //. 

irdiiiif /, Ar, a sin ^, or cos /^, a' sin //, and or' co /< as tin- unknown iiuuiititim, 
the normal r<|iiatioiiK tor ilvtiTiiiiiiing tlicso quantities an- : 

(1) From the ra/uft of /,. 

13 A (if co i w) a sin ; zz A, 
(i'cosiii) A -f- (i'cos* i M) a sin^ = -//, C.IH i 
(i' sin* i M) a ros // zz //, >in / H 

(2) from the rulmx <i( A,. 

13 k -f (i'oos i ) a' cos n' = i'X-j 
(i* cos i ) A -f- (2? cos* i ) a 7 cos ^' = i' k, n .s i n 
I? (sin* i M) a' sin n' =. 2 *, sin / n 

It will l>e observed that all tin- roi-trn-i-nts havini; as n liu-tor either 2 sin i n or 
2 Kin IN C08 i n vanish. 

The value of apparently i not readily determined directly liy least squares: we 
shall therefore a^ume sexeral \alnes of this quantity, and iLM-ertain liy which value the 
conditions can l>et be Hfttintied. The following are the abbreviated values of the purely 
trigonometric summations : 

Mil 6J H 

_ i . / II =Z C 

MM 1 n 

13 sin n -|- sin 13 M 

i cos* n zz = c, 

2 sin 



2 sin a 
K we solve the preceding equation*, and put, lor brevity. 

C - C 
- 



13 c, * 



the resulting expressions for the unknown quantities are: 

A = ''.-A* CEfAjCtti in 
-in ^ = r^ A. C'^'AjCo* in 

a cos ^ zz 2" A, sin 
*i 

A zz CiSkt C2f*,cos i 



a'sin X = -- 



22 



The period of h and k lies probably between fifteen and twenty years, which would 
make the value of n, or the annual motion of the inequality, lie between 18 and 24. 
The following are the values of the various quantities depending on n for the different 
values of n between these limits : 



n 


logc 


log a 


log s t 


logC 


logC 


log C a 


18 


0.756 


0.715 


0.893 


9.213 


9.172 


9-571 


1 9 


0.705 


0.707 


0.898 


9.097 


9.099 


9.506 


20 


0.644 


0.705 


0.900 


8.977 


9.038 


9-447 


21 


0.577 


0.709 


0.897 


8.858 


8.990 


9-395 


22 


0.498 


0.718 


0.891 


8.734 


8.954 


9-350 


23 


o 406 


0.731 


0.882 


8.604 


8.929 


9.312 


24 


0.291 


0.747 


0.870 


8.453 


8.909 


9.276 


25 


0.143 


0.765 


0.856 


8.275 


8.897 


9.246 



n 


2 hi sin in 


2 hi cos i n 


/t ( sin in 


2/6( cos in 




IS 


+ 11.48 


+ 1.96 


- 4.66 


4.66 


'9 


+ n.66 


+ 1.52 


4.68 


5-04 


20 


+ 11.78 


4- 1.09 


- 4-(>9 


5-40 


21 


+ 11.83 


+ o 68 


4.68 


- 5-73 


22 


+ II. Si 


+ 0.29 


4.66 


6.04 


23 


+ U-73 


0.08 


4.62 


- 6-33 


24 


+ 11.58 


0.44 


4-57 


- 6.60 


25 


t- H-37 


0.78 


4-50 


- 6.86 



The preceding equations now give the following separate values of the unknown 
quantities, corresponding to the various assumed values of n: 



n 


A 


a 


/< 


k 


a' 


f' 




18 


O.72 


n 
'53 


164.0 


o.73 


1.81 


160.8 


'9 


0.69 


1-53 


165.2 


0.61 


1.71 


'59-7 


20 


0.66 


i-53 


166.3 


0.49 


1.62 


158-5 


21 


0.63 


1-54 


167.2 


0.39 


1-53 


157-2 


22 


0.61 


-55 


168.1 


0.31 


1.47 


156.0 


23 


0.60 


'-57 


169.0 


0.23 


1.42 


'54-8 


24 


0.58 


1-59 


169.8 


0.17 


1-39 


'53-6 


25 


0.56 


1.61 


170.4 


O.II 


1.36 


152.6 



There can be little serious doubt that in the case of the present inequality the 
theoretical values of // and n' should oe the same; and it is also probable that those of 
a. and a! may be substantially identical. The small differences between the values of a 
and a' and of /* and //' add so much weight to this probability that we shall make 



another solution o!' tin- equations on ill.- Miii|>..Mtion I hat a a anil /*' = p. The nor- 
mal ci|iiations then lu-co : 

13 A ca sin p =. 2/i, 

ch+ 1 3 nr sin ;< = 2?A< cos in 2k { sin i n = 5, 
13*+ coco* n i'A, 

rA:-f 13 orcos// = 2 k i cwin 2h i s\nin = S t 
The solution of these equations is: 



, ___ 

'3 ^ "3 < 



"'3*-? if--* 

A coni|Hirison of the separate solutions ol the equations in A and k shows that the 
value of n which best satisfies the conditions lies between .22 and 25. The values 
of A, k, a, and ft were therefore derived only from the lost equations for the last four 
values of M. For each of these separate values of n, the corres|M>ndtng values of A< and 
4j were computed from the formula* 

A ( rr A a sin (ft -\- i n) 
kf == k -f- cos (ft -j- i n) 

in which, it will be remembered, the index / is simply the number of the year from 
1868 ; so that we have, 

For 1862, ' = 6 

For 1863, i = -5 

etc., etc. 

These computed values of A, and k, were then compared with the values derived 
directly from observations, and given on |>uge 20. and the sum ol the squares ot the out- 
standing residuals was taken. The values of the unknown quantities, together with 
the sum of the squares of the residuals, are a.s follow : 






4 


i 





f 


Z 




JJ 


+ 0.66 


+ O.J4 


t 
'54 



161.2 




3.oj 


*3 


+ 0.63 


+ o.7 


5 


161.3 


J-iTO 


*4 


+ 0.61 


+ o.*o 


1.51 


|6|.$ 


34 


*$ 


+ 0.58 


+ 0.14 


-49 


161.7 


J 441 



The sum of the squares becomes a minimum for n = 22. 8. showing a period ol 
the inequality of I5 7 .8, with a possible error of a \ e.ir or more. The formula- for A, ami 

ki thus become : 

A, = + o".64 i".52 sin (i6i.2 + 22.8 i) 

kt = + o".28 + i".52 cos (l6i.2 + 22.8 i) 



from which we have the following comparison of the computed and observed values of 
h i and k t : 



Year. 


fc 





C. 


0. 


O.-C. 


C. 


O. 


O.-C. 


1862 


4- O.OI 


+ 0.04 


+ 0.03 


4- 1.67 


+ 1.23 


0.44 


1863 


0.48 


0.64 


0.16 


4- 1.32 


+ 1.78 


+ 0.46 


1864 


- 0.79 


- 1.07 


0.28 


+ 0.80 


+ 1.09 


4- 0.29 


1865 


- 0.88 


1.03 


- 0.15 


+ O.22 


- 0.15 


- 0.37 


1866 


- 0.74 


- 0.47 


+ 0.27 


- 0.38 


+ O.IO 


4 0.48 


1867 


- 0.37 


- 0.93 


0.56 


- 0.85 


0.36 


+ 0.49 


1868 


4- 0.14 


+ 0.34 


4- O.2O 


- 1.16 


- 1.46 


0.30 


1869 


+ 0.74 


+ 1.67 


+ 0.93 


- 1.23 


- 1.56 


- -?3 


1870 


+ 1-33 


4 1.48 


+ 0.15 


- 1.07 


1. 14 


0.07 


1871 


+ 1. 80 


+ 1.65 


- 0.15 


0.70 


0.36 


+ 0.34 


1872 


4 2.09 


+ 2.15 


4- O.O6 


o. 18 


0.12 


+ 0.06 


1873 


4- 2.J5 


+ 1.91 


O.24 


+ 0.42 


f- o. 16 


0.26 


187; 


+ I.gS 


+ 1.92 


O.O6 


+ I.OO 


4- 0.60 


0.40 



The probable residual for each year is o".27. 

We have supposed the hypothetical inequality of longitude to be of the form 

4v ^ sin g -f- k t cos g. 

Substituting in this the periodic part of h t and k^ and replacing i by /, which now repre- 
sents the time in years from 1868.5, it becomes: 

or 

Jv= i". 5 2 sin [g + 22.8 (Y- 1857.5)] 

The entirely unexpected character of the periodic term thus brought to light ren- 
ders its verification by a longer series of observations very desirable. For this purpose, 
we need comparisons of observations 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 Greenwich Observations for 1859 contain, as an appendix, a 
comparison of the longitudes and latitudes from Hansen's tables with Greenwich observa- 
tions from 1847 to 1858 inclusive ; and I have utilized the comparison of the longitudes 
derived from meridian observations in the following way : 

A list of limiting dates to tenths of a day was made out, including the whole twelve 
years, and showing between what dates the moon's mean anomaly was found in each 
sextant. The sum of the errors in longitude given by the meridian observations 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 applied, for the reason that 
most of them could be treated as accidental errors, and the means could be taken so as 
nearly to eliminate the effects of the larger ones. A specimen of the form chosen is 
here given. Under each of the several values of g, given at the tops of the several 



2f> 



columns, is .shown, f,r*tly. th,- .l a ti> at which - lia.l that particular value: ami. Moomlly, 
the MIL i o! tli.- n-si.luals in longitude .lurin- the- period of 4^.6 l.elweei, tliat dale mid 
this one next following together with the number of the residuals, the latter I.ein:/ in 
small subscript (inures. 



,.*. 


/=*' + 


/=!*+ / = l8o' + 


/ = 140' + 


/- **>' + 


1847- 
Jan. 19. D- 1.9, 


1847. 
Jan. 14.1- 3.1, 


847. " 
Jan. 1.1+ i j, 


Jan. S.I . . 


1847 
Jan. 10.4+ 1.9, 


1847- " 
Jan. 15.0 . . 


Feb. 16.1 i.o, 


Feb. to. 7 + 0.4, 


Jan. 18.8+ 3. i ( 


Feb.. 1.4+ 3.7, 


Feb. 7.0+ 5.6, 


Feb. 11.6 . . 


Mar. 15.7 . M 10.3- 3.0, 


Feb. 15.34- 4 S 


Mar. 1.9+ 3.7, 


Mar. 6.5+ 2.3, 


Mar. ll.l . . 


April 11.3 . . I April 16.9 . . 


Mar. 24 94- 6.1, 


Mar. lu.s- 0.4, 


Apiil 3.1 


April 7.7+ 3.1, 


May 9.8 . . 


May 14.4 . . 


April ai. sf 3.1, 


April 16.0+ 1.7, 


April 30.6 . . 


May $.14- l.o, 


June 6.2+ 1.8, 


June 10.8 . . 


May 19.0+ 1.8, 


May 13.6+ 3.5, 


May 18.1 0.3, 


June 1.8+ 4.1, 


July 3.8+ 4.4, 


July 8.4 . . 


June 15.4- 1.4, 


June 20.0 0.6, 


June 14.6 i.i. 


June 19.1+ 1.9, 


Aug, 0.4- 0.3, 


Aug. 5.0 . . 


July 13.0 . . 


July 17.6- 1.6, 


July 11.1+ 1.9, 


July 26.84- 6.8, 


Aug. 18.04- 5.9, 


Sept. 1.6+ 3.8, 


Aug. 9.6 . . 


Aug. 14.1 . . 


Aug. 18.8 8.4, 


Aug. 13.4+ 8.1, 


Sept. 24.64-11.1, 


Sept. 19.1 


Sepl. 6.1 


Sept. 10.8 . . 


Sept. 15. 4f 3.6, 


Sepl. 20.0+ i.o, 


Oct. M. 14-12.14 


Oct. 16.7 f 8.7, 


Oct. 3.8 . . 


Oct. 8.4 . . 


Oct. 13.0 . . 


Oct. 17.6- 7.3, 


Nov. 19.7- 1.2, 


Nov. 13.34- 9.3, 


Oct. 31.34- l.o, 


Nov. 4.9 . . 


Nov. 9.5 


Nov. 14.1 o.o, 


Dec. 16.1- 3.4, 


Dec. 10.7 


Nov. 17.6+11.4! 


Dec. 1.1 . . 


Dec. 6.8 . . 


Dec. 11.4- 1.6, 






Dec. 15.3+ 0.7, 


Dec. 29.9+ 0.7, 


De 34.$ 


Dec. 39.1- 1.8, 


1848. 


1848. 


1848. 


1848. 


1848. 


1848. 


Jan. 11.7- 7.3, 


Jan. 17.3 . . 


Jan. 21.9 . . 


fan. 26. $4 o.i, 


Jan. 31.1 . . 


Feb. 4.7 . . 


Feb. 9.3- 8.1, 


Feb. 13.9 6.1 


Feb. 18.5- 1.4, 


Feb. 23.1- 1.4, 


Feb. 17.7 0.8, 


Mar 33 . . 


Mar. 7.9- 1.8, 


Mar. 11.5- 4.3, 


Mar. 17.1+ 4.7, 


Mar. 21.7 . . 


Mar. 16.3 . . 


Mar. 30.9 . . 


April 4.5 . . 


April 9.1- 4.1, 


April 13.7- 1.5, 


April 18.3- 1.8, 


April 11.9+ 1.9, 


April 27.5 . . i 


May 2.0 . . 


May 6.6- 7.4. 


May II. if i.o, 


May 15.84 1.4, 


May 10.4+ 9.0, 


May 25.0 . . 


May 29. ft . . 


June 3.1 8.9, 


lunc 7.8 0.9, 


June 12.4 o.l, 


June 17.0 . . June 11.6+ 2 i, 


June 26.2 . . 


June 30.8 2.6, 


July 5.4- 0.6, 


July 10.0- 4.8, 


July 14. 6 f to. 4, 


July 19.1- O.I, 


July 23.7 . . 


July 28.3 . . 


Aug 1.9- 5.4, 


Aug. 6.$- 3.1, 


Aug. ll.l 


Aug. 15.74-17.4, 


Aug. 20.3+ 1.2, 


Aug. 14.9 . . 


Aug. 19.5 . . 


Sept. 3.1- 6.7, 


Sept. 7.7- 5.1. 


Sept. 11.3+IS 7i 


Sepl. 16.9422.$. 


Sepl. ll.$ 


Sept. 16. i 


Sepl. 30.7- i.o, 


Orl. $.3+ 1.0, 


Oct. 9.8+ 83, 


Oct. 14.4+ $.1, 


Oct. 19.0 ... 


Oct. 13.6 . . 


Oct. 18.1 . . 


Nov. i.8f 1.9, 


Nov. 6.4 4.9, 


Nov. 11. o* 6.9, 


Nov. i$. 6412. ft, 


Nov. 10.1 


Nov. 14.8 . . 


Nov. 19.4- 9.4, 


Dee. 4.0- 7.1, 


Dec. 8.$- 6.5, 


Dec. 13.1+ 5.1, 


Dec. 17. 7f 7.0, 


Occ. 22.3 . . 


Dec. 16.9 . . 


Dee. 31.5- $.71 







If we follow any one of these vi-rtical mluimis, we -Indl timl that the date* corre 
s|Niiul successively to all points of tin* lunation in a |ieriod of 412 d.i\ The first 
observations of each period will be tin- last in-s uf tin- lunation, an- 1 tin- la>t on.-> tli..^,- 
made immediately alter new moon. llHurrn i-acli pair o|' pnioiU will \- a nap. L'''u- 
.-tally of three or four months, during which tin- moon was. at tin- COfTOponding |Mimt> 
of mean anomaly, too near the sun to be obsrm-d. Il tin- oli>-i \alions are equally 
scattered through each period, all the rrrors arisinir from rrnirx-ous -i-iiii-diaiii-t)-r and 
parallactic inequality will lc eliminated. The umrral minuteness of tln-sr rrmrs. and 
their approach to a lialance durinc each ol tin- pi-ritnU in ijiu-tion. an- such as lo render 
them insiunilicnnt, if we take the mean roults, not ly years, but l>y |ierio.l- This i< 
the course ailnpleil : the partial period* at the lwginnii;i; ami end of the entire seriefl of 
olix,-r\ations In-ill^' oniiti.-.l The first period actually employed was tli 



26 



to the sextant 24o-3OO, in which the first observation was made on January 10, 
1847, and the last on September 18 of the same year. The last period corresponded 
to the sextant, i8o-24o, the last observation in which was on November 13, 1858. 
There were, in all, ten periods corresponding to each sextant, and hence ten sets of 
equations, each giving mean values of h, k, and SI for periods extending through a little 
more than a year. Each residual gave an equation of condition, for the coefficients of 
which the mean value corresponding to the entire sextant was taken. These values for 
the several sextants are as follow : 



I 


g 


sin,? 


COSf 


sin'- 


s\ngcosg 


cosV 


I 


0-60 


+ 0.48 


+ 0.83 


0.23 


+ 0.40 


0.69 


2 


60- 120 


+ 0.96 


o.co 


O.QI 


o.oo 


o.oo 


3 


120- 180 


4- 0.48 


- 0.83 


0.23 


- 0.40 


0.69 


4 


180-240 


- 0.48 


0.83 


0.23 


+ 0.40 


0.69 


5 


240 - 300 


0.96 


o.oo 


0.91 


o.oo 


o.oo 


6 


300-360 


- 0.48 


4- 0.83 


0.23 


- 0.40 


0.69 



The sums of the residual errors, corresponding to each period and each sextant 
arranged in chronological order, together with the number of residuals of which each 
sum is formed, are as follow: 



Mean 
date. 


' = 5 


1 = 6 


>' = I 


1=2 


= 3 


1=4 


1847.8 


+ 6.5 


4- 14-4*1 


4- l5-4w 


- ii.7w 


4- Q.O.J, 


- 16.717 


1848.9 


+ 6.7 


+ 8.7 


- 33-087 


I.Qu 


4- 23.2,8 


4- 3'.5.7 


1850.1 


+ . 5 


- 34-i7 


- 40.9*) 


- 9.1.1. 


4- 22. 2 M 


4- 33- Q.". 


1851.2 


4-5 3 


- 59-4i'i 


- 50.7m 


- 23-5*1 


4.8.1 


4- 21- J. , 


1852.4 


- 42.8 a 


- 5. ON 


- 48. OIK 


- 2i.5, 


+ 35-0.;,, 


4- 25.4..:, 


I853-5 


- 31-2*1 


-106.9,5, 


63-631 


+ 1.2,, 


4- 6.0,1 


- 38. o 


1854.6 


- 30.317 


- 94 (> 


- 35 -4 


4- 4-28 


4- I.7n 


- 24. 4m 


1855.8 


- 24.3n 


SO.OIB 


7-3*1 


6.9,., 


22.8| 


- 4'-i" 


1856.9 


- 36.2 


- 23.8 1B 


+ 15. 4u 


4- 4-2ar, 


- 48. 


- 77-Oi7 


1858.1 


- 54-9-w 


- 48.91-, 


- 56.7^1 


- 47-611. 


- 76.9 


- 46.2,8 



The dates given in the left-hand column are those corresponding to the mean of 
each horizontal line. 

Patting .<?,- for the mean value of sin g corresponding to the index i, as already 
given; r^ for that of cos <r; and W; for the corresponding number of observations, the 
normal equations are: 



h 



,. r,.) Jl 



(2 Wi .v, e { ) k = 2 * 
(2 n t c,. a ) k=2c t 



I lie values of // and k thus given by the normal equations formed from the system 
of residuals shown in each horizontal line are shown in the next table, which also shows 



87 

tin- u. i\ 111 wliirh they are treated. Fur tin- sake ol completeness, (In- corie.s|N>nding 
quantities already l,,im,| for (In- peiii.d iSf>j-;j are added, and included in (In- discus 
i, \\hicli nw proceeds as follows; tin- method adopted l.rin^ on.- wliicli. though li'M 
Tons than the former inn-, will chow in a stronger liylit (lie e\ idence on which the 
new inequality depends. 

lie Imsis of tin: discussion, we take thr independent values ol A and /r, derived 
from each series )f observations, which values an- uiv<-n in tin- second and third columns 
nl' the tahlr. A preliminary coni|>nriMMi <>r tin- first (WMI of film* (1847-58) with 
the values uf h anil k derived from the formula- aln-ady -.'ivi-n imlical. > a diminution of 
the constant terms of those quantities, so that, instead of -f o".04 and + o".28, thej 
liecome, iw a first approximation, 

/. = + o". 5 o 
^ = + o".io 

TheM* constants are now ultracted from tin- \alue> of /; and /.. I.MMIIL: a ^. m x ,,i 
residuals i{i veil in the fourth and fifth columns, which, if (In- |n..,lir term under in- 
vestigation has no existence, should ) n-_'.n.|. .1 .1- dm: to errors of oliservatimi, and, in 
the contrary case, should le rapreMntaUfl liy the formula' 

h' = a sin (/i -f nt) -f accidental errors 
k a cos (p -f *0 + accidental errors 

To show clearly how far they are thus represented, we determine a coefficient, a, 
and an angle, .V, by the equations 

a sin .V = - //' 
acoA r = k' 

The next two columns give the se\end \alue> of or and A* thus obtained. The 
nearly regular progression of the niiijl' -V is too striking to lie overlooked. To see how 
nearly this angle can be represented as one increasing uniformly with the time, \\e so|*e 
the necessary eqimlions of condition by least sipmres. It is obvious that the urealer tin- 
value of a the more certain will be the value of .V: we therefore uive weights propor- 
tional to or. Moreover, weights nearly twice as great in proportion an- ^'iven to the 
second series (1862-74) as containing the results from two observatories, and beinj; 
more carefully corrected. The values of // and // thus obtained by the method of 1< 
>'l'iares are : 

//=|6 4 .64 A 

H= 20 .8 .47 

The probable error of a value of .V of weight unity comes out 

33 

The residuals still outstanding are shown in the column _M This value of H \ 
2 less than that found from the second series of observations alone, and an examination 
of the residuals shows that there is a real discordance between the \alues of (he angular 
motion of .V sjivcn by the two series. It is quite likely that the relative weights assigned 



28 



to the older series of observations are twice as great as they should l>r, and that t In- 
most probable value of the angle A r lies nearly halt-way between the two values 



and 



22 .8<>- 1868.5) 

20.8(/- 1868.5) 



found from the last series alone, and from the two combined. 1 judge that the most 
probable value is . 

.2 + 2i.6 (t 1868.5), 



and that the probable error of the annual motion is more than half a degree, but less 
than a degree. The column 4' N shows the residuals given by this value of N. 



Mean 
date. 


A 


, 


* 


*' 


a 


N 


Wt. 


/' + t 


A AT 


A' A" 


1847.8 


0.08 


II 

+ 0.55 


It 

- 0.58 


N 

4 0.45 


n 
0.74 


. 

52 


I 


94 


4 42 


4 24 


1848.9 


- 0.55 


- 1.33 


- 1.05 


- 1.48 


1.82 


'45 


3 


118 


- 27 


- 44 


1850.1 


O.2O 


- I.QI 


0.70 


2.01 


2.13 


161 


3 


Mi 


20 


- 36 


1851.2 


- 0.32 


- 1.92 


0.82 


2. 02 


2.18 


158 


3 


I6S 


4- 7 


- 8 


1852.4 


+ 0.26 


- 2-45 


- 0.24 


- 2.55 


2.56 


175 


4 


189 


+ 14 


o 


1853.5 


+ I. 10 


- 1.88 


4- O.6o 


I. .98 


2.07 


197 


3 


212 


+ 15 


4- 2 


1854.6 


4- 1.45 


- 1.40 


4 0.95 


- 1.50 


1.77 


212 


3 


236 


4 24 


+ 12 


1855.8 


+ 0.77 


+ 0.31 


+ 0.27 


+ O.2I 


0-34 


308 


i 


26O 


- 48 


- 60 


1856.9 


+ 1. 7<> 


+ 1.82 


4-1.26 


4 1.72 


2.13 


328 


3 


28 4 


- 44 


- 55 


1858.1 


- 0.17 


+ 0.66 


0.67 


+ 0.56 


0.88 


5 


I 


37 


-103 


112 


1862.5 


+ 0.04 


+ 1.23 


- 0.46 


4- 1.13 


1.22 


22 


3 


40 


4 18 


4 12 


1863.5 


- 0.64 


+ 1.78 


- 1.14 


+ 1.68 


2.03 


34 


5 


f)I 


+ 27 


4 21 


1864.5 


- 1.07 


4- 1.09 


- '-57 


4- 0.99 


1.8 5 


58 


5 


Si 


4- 23 


+ 19 


1865.5 


- 1.03 


- 0.15 


- 1-53 


0.25 


'55 


99 


4 


1 02 


+ 3 


I 


1866.5 


- 0.47 


+- O.IO 


- 0.97 


o.oo 


0.97 


90 


2 


123 


4 33 


4- 30 


1867.5 


- 0-93 


0.36 


- 1-43 


0.46 


1.50 


1 08 


4 


144 


4 36 


4 34 


1868.5 


+ 0.34 


- 1.46 


0.16 


- 1.56 


1.57 


'74 


4 


165 


- 9 


- II 


1869.5 


+ 1.67 


- 1.56 


4- I.I7 


- 1.66 


2.03 


215 


5 


185 


- 30 


- 30 


1870.5 


4 1.48 


- 1.14 


4 0.98 


- 1.24 


1.58 


218 


5 


206 


- 12 


12 


1871.5 


4- 1.65 


0.36 


4- 1. 15 


- 0.46 


1.24 


248 


3 


227 


21 


20 


1872.5 


+ 2.15 


- 0.12 


4- 1.65 


O.22 


1.66 


262 


4 


248 


- '4 


- 12 


>873.5 


+ 1.91 


+ 0.16 


4- 1.41 


+ O.O6 


1.41 


272 


4 


269 


- 3 


I 


1874.5 


+ 1.92 


+ 0.60 


+ I.4 


4 0.50 


1.50 


289 


4 


289 


o 


4- 4 



The old and new series of observations agree well in giving for the value of tin; 

coefficient of this term, 

The, old series, at i". 66 

The new series, a i"-55 

The effect of the accidental errors will be, on the whole, to increase the value of 
the Coefficient. I consider then-litre that the value 

=i".5o 



89 

m.i\ In- .lilp|>tet| ;i.s (In- niil |>i..l..ilili- \\hirli . MII lie deliteil Itolil all ill.' obxen .(I Mill- 

It u. Militr.irl. from each valiir of li ami k in (lie precedm;.' table. tin- p. ii...|i. 

portiuiu 

/*' = - i".50 hiti [i6j.2 + 2i.6 (I 1868.5)] 
^= i".soK[i63 .2-f 2i.6(/- 1868.5)] 

ami take tin- mean value of the ootttanding remainder fur each series of observations 
we tiinl it to lie as follows: 

( )ld *cries, A. = + o".33 ; *. = - o". i 7 
New series. A, = -f- o".65 ; A, = -j- o"-36 

The differences, o".oi and o".o8, lictween these last valued ami those found on page 
23 arise from the different value, of the |>eriodic term. I consider that the result* of the 
second series are entitled to three times the weight of those of the first, and shall there- 
fore put for the definitive values of It and /., 

A = + o". 5 7 + A' 
*- + o".23-M' 
The correnpoodilig correetions to the eccentricity and longitude of |NTigee are: 



6r = + 2.2 

The corrections to the IIHMHI'- longitude arc: 
5/-c= h sin g k cos g 

o".57 sin g o".23 co^ + i".5o gin (g + JV 90). 

The last term U the hitherto-unsuspected inequality indicated by observations, but not 
yet known U* be given by theory. It may be either an inequality of the eccentricity and 
perigee having a period of about i6j) years, or one of the moon's mean longitude having 
a period of 



Substituting first for A', and then for g, their values in terms of the time, the expres- 
sion for the inequality of longitude bccomen 

i".508in [g+ 73-2 + ai.6( 1868.5)] = i".5own (s6.8 -f 13.! 2413 r), 



r being the time in days counted from Greenwich mean noon of 1850, Jan. o. 

It would perhaps be. premature to introduce so purely empirical a term as this 
into lunar tables for permanent use; but where, as at present, it is requisite to obtain the 
corrections to the (aides during a limited period with all pissild- accuracy, tin- evidence 
in favor of the reality of the term seem> >tiun^ enough to justify its introduction. The 
only apparent cause to which the term can be attributed is tin- attraction of MHIH- one 
of the planets. 

In the investigation of corrections to the longitude, it only remains to determine 
the slowly-varying corrections to the mean longitude, or to /<*:, -riven by the ..!.- 
timis To determine the errors of short period, we have applied several emu dion- t 
the residuals, not as real, but only to render the various observations comparable U 



30 



have now to consider the pure results of observations as Ihey would have been had these 
corrections not been applied. These for the second series of observations are found 
by taking the sum of (i) the mean of the small corrections, applied on account of 
observatory and limb, to compensate for the systematic differences between results from 
different limbs or different observatories; (2) general corrections to make the residuals 
in the mean very small ; (3) remaining outstanding correction found by solving th ( . 
equations of condition. 

The corrections from both series are as follow: the corrections since 1862 may- 
be very closely represented by a term increasing uniformly with the time, as is shown 

by the last two columns. 

First series. 



Date. 


i'z 


Dale. 


*6s 


1847.8 


- 0.15 


I853-5 


-* 1-77 


1848.9 


- 0.43 


1854.6 


+ 1 . 40 


1850.1 


+ 0.32 


1855.8 


+ 1.24 


1851.2 


+ 1. 13 


1856.9 


+ 1.50 


1852.4 


+ 0.93 


1858.1 


+ 2.40 



Second series. 



Year. 


(') 


(2) 


(3) 


ntz 


a + lit 


A 


1862.5 


+ "-45 


+ 2. 10 


i 0.04 


+ 2.59 


+ 1.52 


+ 1.07 


1863.5 


+ 0.45 


-+- 1.20 


- 0.27 


1-1.38 


+ 0.60 


+ 0.78 


I $64 . 5 


o.oo 


O.OO 


- 0.49 


- 0.49 


0.32 


- 0.17 


1865.5 


- 0.15 


- 1.15 


0.62 


- 1 .92 


- 1.24 


- 0.68 


1866.5 


- 0.15 


2.00 


- 0.75 


2.90 


- 2.16 


- 0.74 


1867.5 


- 0.15 


- 3.40 


0.41 


- 3-96 


- 3.08 


- 0.88 


1868.5 


- 0.15 


- 4-05 


O.2O 


- 4.40 


- 4.00 


0.40 


1869.5 


+ 0.08 


- 4.85 


- O.2I 


- 4.98 


- 4.92 


0.06 


1870.5 


+ 0.08 


- 5.50 


0.09 


- 5.51 


- 5.84 


+ o.?3 


1871.5 


o.oo 


- 6-35 


- 0.52 


- 6.87 


- 6.76 


- O.II 


1872.5 


- 0.15 


- 7.25 


0.22 


- 7.62 


- 7.68 


+ 0.06 


1873-5 


o.oo 


- 8.30 


+ 0.10 


8.20 


- 8.60 


+ 0.40 


1874.5 


o.oo 


- 9-45 


+ 0.38 


- 9.07 


- 9-52 


+ 0.45 



IN\ I >IK.\II(iN cl I UK I'ol.AK DIM \M I AM> I.AII II HI 

It i- .1 singular circumstance that (luring the last six years, at least, the ohsn \.i 
tiiin> nl tin- inn m's pular distance are much less accurate than those of its right asccn- 
simi. Whether this is to he attributed to the instruments, or whether it in n result of 
irregularities in the outline of the lunar globe in the polar regions, cannot at pres- 
ent \H* decided. T*t whatever cause we attribute the errors, their existence renders a 
rigorous treatment ol the individual observations of little value. We shall therefore, 
from the whole of the errors in declination, geek to obtain the liest corrections to the 
inclination ami node of the moon's orbit. 

From the derivatives of the moon's declination relatively to its true longitude, the 
inclination, and the node, which have already been given, we obtain: 

tt d& 3ti i ^ jr/i i d& 
do ,. <w-f- 00 -4- 

dl r dO r di 

61 l>eiiig known from the data already given, the equations ol condition will be 
thrown into the form 

'^ m i *W . , dA t . 
i AO 4- -- dt AS Al 
idtt itt dl 



From the numerical expressions already given, we hav 



t j dt = sec A [(0.40 -f 0.08 cos 0) cos / + 0.08 sin sin /] */ 
dl 

If we put 

fi\ tin- correction to the moon's mean longitude, 
K 0.40 + o.oS cos 0, 
II =. o.oS sin 0, 

we have the quantities of the first order, with respect to the i-ccentricilies, 
[A" cos / -f- //sin f] [l -j- 2 rcos (\ IT) ] wed 

(I A. 

The largest terms in sec 6 arc 

1.040-!- .016 cos .040 cos 2 A .016 ! ' .' A 
while, if we replace / by the mean longitude, A, \\e skill h:i\. 

/ = A -f 2 e sin (A w) 

sin / = sin A -f- f ' (2 A w) r sin ir 
cos / = co A + e co (2 A JT) e con JT 

If we substitute these various quantities in the expression for SI, we shall find 



32 

ho sensible terms depending en the sine or cosine of the arguiiu'iit of latitude, A 0. If 

i & 7 / 

we substitute for SI its value in S\ we shall find the principal terms in cos 6 to 

at dk 

be 

K cos A -)- H sin A -(- 3 e, Kcos (2 A ?r) -j- 3 c' 77 sin (2 A TT) 

In consequence of the great number of revolutions of the moon through which the 
observations now under discussion extend, I have considered that all except the first two 
terms might be treated as accidental errors, which would cancel each other during the 
course of the observations. Using for 6\ the mean corrections to the moon's longitude, 
we have the following values of the correction to the declination for those errors of 

longitude : 

Year. Correction. 

// // ( 

1862, -+- 0.9 cos I 0.2 sin / 

1863, + 0.6 o.i 

1864, -o.i o.o 

1865, 0.6 + o. i 

1866, 0.8 o.o 

1867, o.i -o.i 

1868, - 1.4 - 0.2 

1869, - 1.8 -o.i 

1870, - 2.2 - 0.4 

1871, 2.8 - 0.6 

1872, 3.3 -0.6 

1873, -3.8 -0.5 

1874, -4.2 -0.4 

The mean correction to the moon's tabular north-polar distance for each year, from 
observations of each limb at each observatory, was taken with a view of detecting any 
constant error of sufficient magnitude to affect the final results for errors of the node 
and inclination. These means should have been taken after the application of the cor- 
rections just found: actually, however, they are the mean corrections given by the 
observations, tiller applying the following constant corrections to reduce the declinations 
to the same fundamental standard : 

To Greenwich observations of N. P. D. To Washington observations of N. P. D. 

1862-67, -0.4 1862-65, -f 0.5 

1868-74, +0.2 1866-67, -I.I 

1868, - I 2 

1869, 0.6 

I <S 70-7 2, 0.4 

1873-74, o.o 

These corrections are approximately those necessary to reduce the slur-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 the introduction of a new 



enn-tant .if refraction in 1868, while tin- change in tl,e Washington correction in 1866 
corre*|.,.n.!* In (lie introduction of the lame transit circle in plan- ,,( thr ul.l niunil cirri.-. 





Correction to N. 






'. 


Greenwich. 


1 




N. L. 


S. U 


N. U 


S.L. 




X 


M 





M 


1861 


O.I 


o. 


- o.j 


O. 


1663 


+ o. 


0. 


- o.$ 


1. 


1864 


+ 0.4 


- 0. 


+ 0.8 


0. 


1865 


- o.j 


o. 


+ 1.1 


o. 


1866 


- 0.7 


0. 


+ -4 


0. 


1867 


- 0.4 


0. 


+ O.I 


- 1. 


1868 


- 0.7 


1. 


+ o.i 


-1- 0. 


1869 


O.I 


. 


- 0.8 


I. 


1870 


- 0.6 


. 


O.I 


1. 


1871 


o.i 


. 


+ fl.i 


1. 


872 


0.0 


. 


- 0.7 


0. 


73 


- 0.9 


+ . 


+ 1.0 


0. 


t74 








- 1-7 


- <M 



The large residuals of the Washington observations of the south limb led to the 
application of the further systematic correction of -f i".o to all those observations before 
combining them all. The corrections arising from the error of mean longitude were 
then applied, and the outstanding residuals were considered to arise from accidental 
errors and from errors of the inclination and mule. The equations of condition thus 

>mc 

0.92 sec 6 [sin (/ 6) 8i cos (/ ) i 69] 66 
or 

sin (/ 0) 6i cos (/ 6) i 60 1.09 cos 6 X ** 

( >\\ ing i<i the smullness of the final residuals, <$<?, the factor 1.09 cos 6 may be consid- 
ered as a constant, and, in the actual solution, has been put equal to unity. Its mean 
value is more exactly 1.04, and its effect may be obtained by dividing the final results 
I iy this factor. 

The final values of the residuals were then arranged according to the values of 
A 0, 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 corre8|K>nding to 
each interval of 20 in the argument, with the corresponding number of observations 
fur each year, is shown in the following table : 



34 



Sums of errors of the moon's corrected declination, f/iren Inj observations at GreeinrirJ/ 

and Washington. 





1862. 


1863. 


1864. 


1865. 


1866. 


1867. 


1868. 




2(5,5 


N. 


SiM 


N. 


Zdd 


N. 


2(5(5 


N. 


<5(5 


N. 


<M 


N. 


2<W 


N. 





,, 




,, 




II 




H 




a 














o to 20 


- 3-8 


8 


+ 1.3 


3 


+ 4- 


8 


+ 5-4 


9 


+26.7 


ii 


- 2-5 


8 


+ 0.4 


9 


20 to 40 


+ 9-6 


9 


+ 5-8 


7 


- 0.4 


9 


+ 6.0 


7 


+ 2.6 


12 


- 2-3 


9 


6.1 


17 


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 to 80 


o.o 


7 


+ 16.4 


10 


+ 6.6 


8 


+ 8.7 


12 


I.I 


16 


+ 14.5 


ii 


- 0.3 


5 


80 to 100 


+ 8.6 


II 


+ 0.4 


12 


+ 11. 1 


6 


+ 7-9 


II 


+ 5-5 


7 


+ 0.5 


10 


-12.8 


12 


100 to 1 20 


+ 3-2 


7 


+ 8.5 


15 


+ 3.2 


5 


+ 7-4 


7 


I.O 


8 


- 6.1 


6 


+ 0.2 


6 


120 tO 140 


- 9-2 


12 


+ 3-1 


8 


6.1 


8 


+ 0.8 


ii 


+11.9 


14 


-12.4 


8 


- 6.8 


ii 


140 to 160 


- 0-3 


4 


- 4.6 


9 


2.2 


5 


- 9-7 


15 


1.2 


10 


- 7-7 


12 


+ 6.2 


14 


160 tOiiSo 


+ 0.5 


9 


10.4 


6 


IO.4 


12 


+ 0.5 


9 


+ 2.2 


10 


- 8.9 


9 


II. 2 


9 


1 80 to 20O 


- "8.6 


6 


- 5-7 


ii 


- 0.6 


7 


- 5-3 


12 


- 7-7 


6 


-15.2 


'4 


+ 3-1 


10 


2OO to 22O 


22.3 


8 


-II. 6 


10 


+ 4-7 


12 


- 5-4 


9 


- 3-3 


TO 


- 6.8 


14 


-ii. 8 


ii 


22O to 24O 


-14.4 


12 


IO.2 


9 


- 8.8 


10 


I.O 


7 


2.0 


'3 


- 5-9 


12 


IO.O 


13 


240 to 260 


-12.4 


7 


12.3 


9 


- 4-1 


8 


+ 4.6 


ii 


+ 1.2 


9 


- 0.6 


9 


- 9.2 


15 


260 to 280 


- 2.3 


4 


- 3-2 


4 


- 8.5 


8 


+ 1.5 


9 


- 5-3 


9 


+ I.O 


8 


+ I. g 


9 


280 to 300 


2.2 


7 


- 4-3 


8 


- 8.4 


II 


- 4.0 


4 


- 3-5 


9 


-ii. 4 


'3 


o.o 


'3 


300 to 340 


- 7-1 


10 


- 6.2 


10 


4- 9.6 


8 


+ 3-1 


5 


O.I 


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 


9 


- 6.7 


'4 


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 


'54 


49-5 


I 5 6 


-25.4 


159 


25.2 


191 


-93-1 


'75 


-85.9 


203 




+31.2 




+45-8 




+ 56.6 




+75-8 




+ 69.5 




+ 21. 




+ 12.2 






-52.8 




29.2 




+ 7-i 




+ 50.4 




+44-3 




-72.1 




-73-7 








Sums qf errors qf tke WMMN'M corra-/**/ / /imi/ion, (< ( 'niiun. <1 





1869. 




1870 


. 


I8 7 l 





1872 




73 





74 







IM 


N. 


1*1 


N. 


sw 


N. 


: 


N. 


ZM 


N. 


XM 


N. 





" 









H 




a 




** 




M 




o lo so 


7-1 




+ JL 

















4- 7-7 




40 lo to 


4- II. S 
4- 6.4 


7 


O 

4- .6 


10 


- 0.8 


u 


- 1.2 


10 


- S.S 


i 


/S 
- 17.0 


u 


to lo 80 


- S-o 


9 


+ $ 


7 


4-I3-* 


ii 


- s.i 


7 


-IJ.7 


7 


- S-4 


14 


100 10 ISO 


s.o 
- 3-7 


u 
7 


4- .S 
- .S 


9 

IS 


4-13.1 
- 6.3 


9 


S.o 


8 


- 1.0 


9 


- 11.4 


11 


iso to 140 


- II- 4 


9 


+ 4-5 


7 


- 1-9 


S 


4- O.S 


14 


- 1.1 


4 


4- S.I 


6 


I to to 160 


- 15-4 


IS 


-11-7 


u 


- 4-6 




-8.9 


9 


4- 7-4 


5 


- 11.6 


u 


ito to 180 


- $ 


II 


- S-7 


3 


- S-1 


9 


- 4.6 


12 


- 39 


7 


- S-9 




180 |O MO 

200 10 ISO 


5-4 
- 5-4 


9 

4 


0.$ 
-IO.S 


IS 


+ 5-4 
- 6.S 


10 




9 


+ $ 


7 


- 19-3 


12 


33010140 


- 6.6 


6 


I.I 


9 


4- 9-1 


14 


- 4-S 


10 


- -7 


S 


- !$ 


6 


14010 ito 


- 18.4 


IS 


II. I 


8 


+ S-6 


8 


-H3-7 


3 


4-14-8 


11 


4-6 


6 


to 10180 


- 7-7 


7 


-15.4 


IS 


- 6.S 


7 


4- S.S 


u 


4-S0.7 


9 


- l.S 





18010 300 


- ll-4 


13 


10. 1 


5 


+ 3-6 


8 


+ S- 


9 


+ 3-3 


10 


- s-s 





30010320 
3soto3o 

34010360 


+ S-3 
+ 5-7 
o.o 


7 
S 
10 


-10.3 


7 
IS 




8 

9 


- 1-3 

IS.S 


II 
IS 


+ 4-3 
4-14-0 


8 

7 
II 


4- 0.3 
4- 4-0 
- 3-9 


10 
10 

9 




-104-9 


55 


93.6 


166 


-47-1 


153 


-67.4 


167 


-4S.6 


140 


-IS4-3 


169 




+ 35-7 




4-33-I 




+S3-8 




4-SO.S 




+90.7 




4- 14 1 






- to.i 




-$9-5 




- 6.7 




-47- 




_+*_' 




-140.1 





The general'iiTegularity of the residuals in declination is such that no great alvan 
tage would result in a separate solution of the equations for the separate years. Tin- 
sum of the residuals for each 20 of the argument wa* therefore taken during the whole 
thirteen years of observation, with the following result: 



l- 


ZAJ 


N. 


%- 


ZA4 


N. 




olo so 




4- 47- 


i<>3 



18010 soo 


- 62.3 


110 


solo 40 


4- 10.7 


i"S 


SOOIO 330 


- 9S 3 


138 


4010 to 


4- 6.1 


119 


330 U> *40 


- 73 3 


126 


to w 80 


4- is. 3 


1*4 


140 to sto 


- 3. 


137 


8010 too 


4- 34-3 


ill 


36010 380 


- 23.8 


106 


1 looioian 


- 36.1 


ill 


380 to 300 


- so.i 


1*3 


i iso to 140 


- 7-4 


iso 


300 to 320 


- 5-4 


S 


14010 ito 


- 65.3 


U4 


jso to 340 


4- 19- 


ISS 


160 to 180 


- 65.4 


IS6 


54010360 


4- SO.S 


no 



36 

Leaving in the equations a constant term Sp, representing the mean constant error 
still outstanding in the measures of declination, the solution of the equations of con- 
dition given by the residuals gives the following results : 

dp o". 1 7 



Correction to the inclination, o".i5 

Correction to the longitude of node, +4"-5 

This correction to the longitude of the node from Hanson's tables implies a dimi- 
nution of the secular retrograde motion of the node, which is quite accordant with the 
results derived from ancient eclipses. Hansen remarks that an increase of 1 2" per cen- 
tury in the longitude of the moon's node will improve the agreement of his tables with 
ancient eclipses;* and, if we suppose the tabular longitude of the node to have been cor- 
rect in 1825, this would imply a correction of + 5". 2 to the longitude of the node 
in 1868. 

* Darlegung, etc., Th. ii, p. 391. 



87 



\i\lMAk\ i.u.i | FACILITATING nil COMPUTATION <>i nil COR EEC 

ini\> TO M.\\-I W8 "TABLES M l.\ I ( \i . ( .i\ IN \-.\ un n <i CEDING D13 

Tin- following is a summary of the corrections to (In- longitude of (hi- moon from 
ll.niM-ii's tables given by tin- preceding discussion. 'I'll.- first >ix t.-rms an- applicable 
to the listurli-.l mean longitude, or "Argument foudumfntal"; tin- remainder to Un- 
true lonjjitu.le; but they may all \tc used as corrections of the "Argument fon<l<imcntal" 
without serious error: 

OIM on account of diminution of the .so//;/ jxinillax. . n 6; = + o".g6 sin l> 

+ o".07. si .. 

-o". 1 3 sin 
On account of hypothesis (ft- mionally set ./Wr), ////// 

the moon's center of grarity Joes not coinci<t> iritli th, 
tcr ofjigure, together irith th<- correction to the. evec- 
tio* resulting from the correction to theecc<-n//i<iti/. . . n 6z = + o".O9sin g" 

-o".338in 2 It 

o".2i sin (2 D g) 
OH account of term accidentally intru>lun<l into 

the taltlix irith a wrong fign .......... 6v = o".62 sin (2 g 4 gf + 2 w 4 u') 

OH account of corntfi<,n to the eccentricity and jn-rigee 

found from observations during 1847-74 ....... do o".tf8\ng o".2^ctwg 

= o".62 sin (g -|- 202 .o) 
Empirical term, neceuary to satisfy observations, 

but not verified by theory ................ + i".5O8in [g -f 2i.6 ( Y 1865.1)] 

Unexplained correction to the mean longitude, changing slowly from year to 

year ....... .................................... See Table I V. 

The deduction of all these terms, except the 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 page 30, in the column n 62, by suppos- 
ing this quantity to vary according to some simple law, which law changes when necessary, 
so as to satisfy the observations within the moan limits of their probable error. An 
examination of Table IV shows, that, from 1848.0 to 1855.5, tn<> correction is sup|nsed 
to increase uniformly at the rate of o".2O per annum. It is then supposed to remain 
constant until nearly 1863.0, a i>eriod during which the observations are not continiio 
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 correction 

- 5"-53 -o".86 (t 1870.0) + o".02 (t 1870.0)* 
The continuance of this correction beyond 1875.0 is, of course, purely conjectural. 

TABLES FOR APPLYING THE PRECEDING CORRECTIO 
Tin- following tables are designed to facilitate the computation of the correction!! 



38 

just given. To avoid the necessity of referring to Hansen's tables, the values of all the 
necessary arguments are given for the years 1850 to 1889 in Tables I to III. 

Table I: the epochs are January o, Greenwich mean noon of common years, and 
January i of leap years. All the arguments increase uniformly by a unit in a day. 

Argument g is the moon's mean anomaly, converted into days by dividing its ex- 
pression in degrees by 13.065. It is equal to Hansen's argument g diminished by 1 5 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 equal to Hansen's argu- 
ment 33 diminished by 30 days, or, which amounts to the same thing, by o d 47. 

Argument A gives the number of days from the time when the angle 

2g 4g' + 2(0 400' 

was last zero. 

Argument B is that of the empirical term indicated by observations, but not given 
by theory. 

Argument u is that of latitude, or the number of days since the mean moon last 
passed her ascending node. 

Tables II and III do not seem to need explanation. In using the former, care must 
be taken to diminish by one day the dates for January and February of leap years. 

Table IV gives the secular corrections to the mean longitude, or to ndz, obtained 
from observations in the manner already described. 

Table V, argument A, gives the correction for the term introduced into the tables 
with a wrong sign, described on page 9. It is properly to be applied to the true longi- 
tude, and is therefore designated as Sv. 

Table VI gives the empirical term, which, so far as is known, may be applied to the 
true longitude. 

Table VII gives the sum of the terms of mean longitude 

+ o".g6 sin D 
o".33 sin 2 D 
-o". 1 3 sin (# + #') 
+ o".09 sin g' 

The sun's mean anomaly, g', having a period of a year, the sum of these terms can 
be expressed as a function of D and the month, and is given in the table for the middle 
of each month, and for each day of D. 

Table VIII gives the sum of the terms of true longitude which depend wholly or 
partly on the moon's mean anomaly, namely: 

+ o".62sin(+202.o) 
+ o".o; sin (D - g) 



o".2i 



The sum of the terms of n dz are to be reduced to corrections of the longitude in 
orbit by multiplication by the factor 



i -f 2 e cos g -f ? e* cos 2 g] 



This factor, less unity, is given in Table IX. 



eunxenience, lii,- unit (it tin- larlor is omitted Imiu tin- tabular number*, no that 

nly necessary to add tin- product f'X <*- i with n 6: and dr to ha\e tin- cor- 
,"ii ni' tin- true longitude in orbit 

'I'h. -si- corrections beiim applied to tin- lonailiide ul' the moon's center I'miii Han- 
I tallies, that longitude may In: regarded as correct, excepting n small correction, 
which ma\ probably he regarded as constant during any one |M>riod not excocdmi: si\ 
luuntlis, ami which may he attrihutcil to tin- adopted position of tlic equinox. It \\ill 
bo best determined from orcultations nt stars observed at points whose longitudes from 
Greenwich arc accurately known by telegraph, and will then be applicable to the 
determination of the longitude of any station from occultationa. 

If the corrections here deduced are applied to the errors of the lunar ephenieris 
deri\ed from meridian observations, it must be remembered that these observations are 
made on tin- 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 observer, the instrument, and the brightness of the sky. For large instru- 
ments, Hanson's semi-diameter is about i too great, even at night 

The sum of all the terms of n <Jr. <Jp, and FX" <f- fr tn * c tables will lie the 
correction of the longitude in orbit. This will not be rigorously the same as the correc- 
tion to the ecliptic longitude. 

Table X gives the small factor (f./) by which the orbit longitude mu-i IK- increased 
or diminished to obtain the ecliptic longitude. This factor may generally be disregarded. 

Table X also gives the data for the correction of the moon'- latitude, namely, ( i ) 
a factor (F. ft) by which the correction of the moon'.- argument of latitude mu*t he 
multiplied; and (2) the term 

6ft t = o".i5 sin u 

arising from the correction to the tabular inclination of the moon's orbit The correc- 
tion of the moon's argument of latitude heing that of her longitude, minux the correction 
of her node, the whole correction to the latitude will be 

<J/? = <5/?, + (F./?)(Ji;-4".5) 

Table XI gives the factors for convortinir corrections of longitude and latitude into 
corrections of right ascension and declination. The formula' are 

S.jfi dv + (v.d)6v +(ft.a) 6ft 
6 . Dec. = <?/*+ (r . <J) <5r + (ft. <J) 6ft 

The side argument is the moon's longitude, and in the ( oeflicients (.<*) and perhaps 
(ft. a) regard must be had to the moon's latitude also. Three columns are therefore 
for latitude, 5, o, and -f 5 respectively. 



40 



As an example of the use of the tables of corrections, we will commence the 
determination of the corrections for September, 1874. We find (he values of the argu- 
ments for September i, from Tables I to III, as follows: 





S 


D 


A 


B 


u 


1874 . . . 
Sept. I . . 
Periods . . 

Arg. Sept. I . 
Arg. Oct. i . 


5-4 
23.6 
27.6 


12. 1 

7.8 


8.0 
1.9 


20. o 
24.6 
-27.4 


1.9 
26.2 
27.2 


1.4 


19.9 


9.9 


17.2 


0.9 


3-8 


20.3 


39-9 \ 
or 7.8 ) 


47-2) 
or 19.8 f 


30.9) 
or 3.7^ 


.0=19.9 
g= 1-4 


D-g* 


:.8. 5 





The tabular numbers are then found as follows, with an argument increasing by 
unity each day. From Table VIII, we take a mean from columns 18 and 19. 



September . 


i. 


2. 


3- 


4- 


5- 


6. 


7- 


8. 




,, 


,, 


H 


II 


,, 





,, 





Table IV ( iz) . 


- 9.11 


- 9.II 


- 9.11 


- 9.11 


- 9.12 


- 9.12 


9. 12 


- 9.12 


V (<fo). . 


-t- 0.40 


+ 0.55 


+ 0.62 


+ o 59 


+ 0.48 


4- 0.28 


4- 0.05 


0.18 


VI (<!z<) . 


- 1.07 


1.29 


- 1.42 


- 1.50 


- 1.48 


- 1.40 


- 1.23 


1. 01 


VII(As). 


1.29 


- 1-25 


I. 12 


- 0.95 


0.76 


0.56 


- 0.37 


O.22 


VIII (Av) . 


0.65 


0.72 


- 0.77 


0.78 


- 0.75 


0.69 


O.6o 


0.48 




-11.72 


U.82 


i i . 80 


-11.75 


u . 63 


-11.49 


11.27 


II. 01 


ntsXF, Table IX 


I. 12 


- 0.93 


- 0.71 


- 0.47 


0.28 


0.06 


+ 0.18 


4- 0.37 


6v 


12.84 


12.75 


12.51 


12.22 


11.91 


11.55 


11.09 


IO.&4 




















I'P 4".5 . . . 


-17-3 


17- a 


-17.0 


-16.7 


16.4 


16.0 


-15.6 


-I5.I 


Table X (F . /?) . 


4- 0.088 


+ 0.082 


+ 0.070 


+ 0.056 


+ 0.038 


+ 0.019 


0.002 


O.022 


X(d/?,). . 


0.03 


0.07 


O.IO 


O.I2 


- 0.14 


- 0.15 


- 0.15 


- 0.14 


(*--4".5)(F.ft- 


- 1-52 


- 1.38 


- 1.19 


- O.Q3 


0.63 


0.30 


4- 0.03 


+ 0.33 


AS . 


1.55 


1.45 


1.29 


I.O5 


0.77 


0.45 


O. 12 


4- 0.19 


































o 


e 





e 


D's longitude . 


46.5 


60.7 


74.6 


88.1 


101.5 


II4.5 


127.4 


140.0 




II 


n 


II 


II 


II 


it 


,/ 


" 


(l+(v.a)){v. . 


-13.08 


-13.49 


-13.84 


-13.81 


-13.39 


12. 6l 


1 1 . 62 


10.67 


UM'V* . . . 


+ 0.47 


4- 0.32 


4- 0.16 


4- O.O2 


0.07 


O.Og 


0.03 


4- 0.06 


d/R j 


12.6 


-13.* 


-13-7 


-I3-8 


-13.5 


-12.7 


-ii. 6 


10.6 




-o".84 


-0-.88 


o". 91 


O*.92 


o"-9O 


-o.85 


-o".78 


-0-.7I 




II 


II 


a 


H 


u 


a 


" 





(v.fyiv . . . 


- 3-70 


2.64 


- I.4I 


O.l8 


+ 1.02 


+ 2.05 


4- 2.84 


4- 3-35 


(l+/S.rf)<!/5 . . 


- 1.50 


- 1.42 


- 1.28 


- 1.05 


- 0.77 


- 0.44 


O.I2 


4- 0.18 


c)Dec 


- 5-2 


- 4-1 


- 2-7 


1.2 


+ o.a 


4- 1.6 


+ 2-7 


+ 3-5 



41 



This < im|iiil.itiini ln> I.-- -n fiintiiitii-.l In 1X75, .F.imi. try ;i. mnl tlir r-j.iili 

-ll.'U II ill till- liil|ii\Mliy t;il. 



Corrections to th /.'/<A<-mero dtriveil fr m //<ivo/\ '/',,/,/,> <>i' th< M'.<i. t 

of each tltiy,from 1874, Srpteiuber i, to 1875, Jnnn,n->i \\. 



Dale. 
Gr. mean 

DOOM. 


Correction to tabular 


in mran 
noon. 


C< 


Lai. 


to Uhula 

R.A. 


r 


Long. 


Ut. 
- 1.6 


R.A. 
-ia.6 


Dec. 

- s.a, 




Dec. 
+ 3-6 


1874. 
Sept. I -n 8 


874- 
Oct. 1 1 


H 

- 7-5 


** 

+ I.I 


- 6.7 


l 


ta.8 1.3 


13. t 


4. i 


7- 


I.o 


6.8 


3 a 


3 

. 


ia.$ 1.3 

l*.t 1.0 


13-7 
13.8 


a. 7 3 

- i.a i 1 14 


6.9 
6.6 


I.O 

0.8 


6.8 
7.0 


a. 7 
a.o 


7 


11.9 

-11.6 
II. 1 


0.8 
- o.$ 

O.I 


3-5 

-It. 7 
tl.6 


+ o.a | 15 

+ 1.6 I 16 

7 I 7 


6.4 


0.7 


7-0 


J 


6.1 i 0.3 


o.a 


a 


IO.6 +0.3 


10.6 


35 | 8 


6.3 


68 


I.O 


9 10. i 0.5 


9.6 


4-1 , 9 


6.4 


O.I 


6.6 


1.8 


to 9.6 0.7 


8.6 


43 o 


6.8 


0.4 


6.6 


a. 5 


1 1 


9.0 + 0.9 


- 7-9 


+ -4 ! 


- 7-5 


- 06 


-69 


- 3-3 


n 


8.3 i.o 


7-3 


.a l] aa 


8.3 


0.9 


7-3 


41 


13 


7.6 j I.I 


6.7 


.8 a 3 


9-3 


1.1 


B.I 


4-7 


M 


6.8 l.o 


6.a 


3 j| 4 


o-4 


3 


9-a 


5-a 


i$ 

16 
17 


6. a ' i.o 


5-9 


8 


as 

. 
7 


11.4 

-ia.4 
13. a 


5 

- -S 
1*4 


10.6 

-la.a 
3-9 


5-a 

- 4.8 
3.8 


5. a 0.7 


55 


1.6 


: 5.0 0.6 


5-5 


I.O 


aS 


3.6 


i.a 


ll.l 


a-3 


'9 


$.1 0.4 


5-8 


*- 0-3 


- 


13.8 


I.O 


5-7 


- 06 





$.4 + o.s 


6.1 


- 0.4 




13.6 


0.6 


5-3 ' i 


H 


6.1 o.o 


- 6.6 


i.a 


3 


-13. a - 0.3 


-14. a 


+ a.6 


7.0 o.a 


7-* 


a. a 


Nov. i 


13.4 + O.I 


13.6 


36 


13 8.1 


o.$ 


7-8 


3-* 


S 


11.4 0.4 


n. t 


43 


4 


9.4 0.8 


8.5 


4-3 


3 


10.5 0.7 


,6 


4-5 


as 


10.6 1.1 


9-3 


5-* 


4 


9.5 0.8 


8.4 


4-5 


a6 


-11.8 - 1.3 


-10.3 


- 5-8 


5 


- 8.J 


+ I.O 


- 7-4 


+ 4-3 


t7 


ia.7 


1.1 


u. S 


6.0 




7.7 i.o 


6.7 


39 


38 


'3-4 


1.6 


IS. 8 


5-7 


7 


7.1 i.o 


6.3 


3-5 


- 


3-7 


$ 


13-9 


4-8 


8 


6.6 


l.o 


6.3 


3 


30 


13.8 


1.4 


4-9 


3-4 




6.4 


I.O 


6.3 


a.6 


Oct. i 


-3-S 


t. a 


-i$.a 


- t.7 


to 


- 6.3 


+ 0.9 


- 6.$ 


+ a.i 


* 


13.0 


0.9 


14.7 


o.t 


n 


6-3 


0.7 


6.9 


5 


3 


ta.a 


o-$ 


13.6 


+ 1-4 


13 


6-5 


0.6 


73 


* 0.8 


4 


it. 5 


O.I 


ta.a 


a. 6 


J 


6.8 


0.4 


7-7 


O.I 


S 


10.6 


+ O.I 


to.8 


3-3 


4 


7- 


+ O.I 


7.8 


I.O 


6 


- 9-9 


+ 0.4 


- 9-4 


+ 3-9 


5 


- 7-4 


O.I 


- 7-8 


- t.8 


7 


9- 


0.6 


1.4 


4-1 


16 


7-7 


0.4 


7-7 


a-7 


1 


7 


O.I 


7-7 


4- 


7 


8.1 


0.6 


7-6 


3-4 


9 


8. a 


I.O 


7-a 


l-a 


18 


8.4 


0.8 


7-6 


4.0 


to 


7-8 


I.O 


6.8 


4.0 


9 


8-9 


I.O 


7-8 


4-5 



I 



42 



Corrections to the EpJiemeris derived from Hansen's Talks of the Moon, etc. Continued. 



Date. 


Correction to tabular 


Date. 


Correction to tabular 


Gr. mean 
noon. 


Long. 


Lat. 


R. A. 


1 

Dec. 


Gr. mean 
noon. 


Long. 


Lat. 


R. A. 


Dec. 


1874. 
Nov. 20 


- 9-4 


- I .2 


- 8.2 


- 4-8 


1874. 
Dec. 27 


[0.8 


+ 0.6 


IO.2 


+ 4-3 


21 


IO.O 


i-3 


9.0 


4.8 


28 


10.4 0.8 


9-5 


4-7 


22 


10.6 


1.3 


10. I 


4-5 


29 


10. I 


I.O 


8.9 


4-9 


23 


II. 2 


1.3 


II. 4 


3-8 


3 


9.6 


1 .2 


8.4 


4-9 


24 


II. 7 


1.2 


12.7 


2.6 


3' 


9.2 


1.2 


8.1 


4-6 












1875. 










25 


12.2 


1.0 


-13.8 


I .2 


Jan. I 


- 8.7 + 1.2 


- 7-9 


+ 4-1 


26 


12.5 


0.6 


14.1 


4- 0.5 


2 


8.2 I.I 


7-9 


3-5 


27 


12.6 


- 0.3 


13.8 


2.0 


3 


7-8 


I .0 


7.'> 


2.9 


28 


12.5 


0.0 


13.0 


3.3 




7-4 


0.9 


7-9 


2.1 


29 


12.1 


+ 0.4 


11.9 


4-3 


5 


7-1 


0.6 


7-8 


1.2 


30 


-II. 7 


+ 0.7 


10.9 


+ 4-9 


6 


- 6. Q 


+ 0.4 


- 7-8 


4- 0.4 


Dec. I 


II. 


0.9 


9.9 


5-1 


7 


6.8 


+ 0.2 


7-7 


- 0.5 


2 


10.3 


I.I 


9.0 


5-i 


8 


6.9 


O.O 


7-5 


"4 


3 


9-4 


1.2 


8.2 


4.7 


9 


7-2 


- 0.3 


7-4 


2.2 


4 


8.5 


I .2 


7.5 


4.2 


10 


7-6 


0.5 


7-3 


3-0 


5 


- 7-7 


+ I.I 


- 7.0 


4- 3-6 


ii 


- 8.0 


- 0.7 


- 7-3 


- 3-7 


6 


7-0 


I.O 


6.8 


3.0 


12 


8.6 


I.O 


7-5 


4-3 


7 


6.4 


0.9 


6.6 


2.3 


13 


9.2 


1.2 


8.0 


4-7 


8 


6.1 


o.S 


6.5 


1.6 


14 


9-7 


1.3 


8.6 


4-9 


9 


6.0 


0.6 


6.7 


0.9 


15 


to. 3 


1.4 


9-5 


4.8 


10 


- 6.1 


+ 0.4 


- 6.9 


+ O.I 


16 


10.8 


- i.3 


10.6 


- 4-3 


ii 


6.4 


+ O.2 


7-' 


- 0.7 


17 


II 2 


1.2 


II. 7 


3-4 


12 


6.8 


0.0 


7-3 


1-5 


18 


ii. 5 


I .0 


12.6 


2.2 


'3 


7-4 


- o-3 


7-5 


2.4 


19 


ii. 6 


0.8 


'3-1 


- 0.8 


14 


8.0 


0.6 


7-6 


3-3 


20 


ir.fi 


0.5 


13.0 


4- 0.8 


15 


- 8.7 


- 0.8 


- 7-8 


- 4.0 


21 


-ii. 5 


O.I 


-12.4 


4- 2.2 


16 


9-3 


i.i 


8.1 


4-7 


22 


ii. I 


+ 0.2 


ii. 4 


3.3 


17 


9.9 


I .2 


8.6 


5.0 


23 


10.7 


0.5 


10.3 


4-1 


18 


10.3 


1.3 


9.2 


5-1 


24 


IO.2 


0.7 


9-3 


4-5 


19 


10.7 


1-4 


IO.O 


4.8 


25 


9.6 


I.O 


8-5 


4.6 


20 


II. 


- 1-3 


II. 


- 4-1 


26 


- 9.1 


+ i.i 


- 7-9 


4- 4-6 


2 


II. 2 


1.2 


12. 


3-1 


27 


8.5 


i.i 


7-5 


4-3 


2 


II. 4 


1.0 


12.7 


1.7 


28 


8.1 


1. 1 


7-2 


4-0 


2 


II. 4 


0.7 


12.9 


0.2 


29 


7-7 


i.i 


7-2 


3-5 


2 


II. 4 


0.4 


12.6 


+ 1-3 


30 


7-5 


I.O 


7-4 


2.9 


2 


-11.3 


O.I 


12.0 


+ 2.6 


3' 


- 7-4 


+ 0.9 


- 7.7 


4- 2.3 


2 


II. I 


+ 0.3 


II. I 


3-6 













TABLES. 



T.\ I; I. I.S. 



TAIH.I 1 


TM...K II 


/ \ilutt i/ tkt Ar^umcnti for tkt /. 


KtifuftitiH nf Ikf Arguments la tkf tent- 


Hing of fafk i 


itey of fiit A month. 




/> 




H 


M 


lib. 


t 


D 


A 


/ 



























iSso 


1.8 


16.7 


-J 43 


5 


Jan. > 


o.o 


o.o 


o.o 


o.o 


o.o 


1851 


8.6 


7-3 


o.o 12.7 


9-S 


Feb. o* 


3.4 


1-S 


14-9 3-6 


38 


iSjt B 


16.4 


9-4 


10.9 aa.i 


ai.7 


Mar. o 


3-9 


0.0 


10.6 4.1 


4-6 


1853 


3.l 


20.0 


4-7 


3 


5-8 


April o 


7-3 


1.4 


9-J 


7-7 


8.4 


8$4 


-4 


l.'l 


U-6 


u .5 


17-0 


May o 


9.8 


1. 


7-0 


10.3 


11.2 


llM 


9m 


11.8 







| _o 














055 

1856 B 




17.0 


t3-4 


* 
3- 


"I- V 
9 


"3- 


July o 


S-7 


3- 


,, 




"7-7 


857 


3 8 


4-S 


13.1 


10.3 


<4- 


AUK. o 


19.1 


r. 


i.a 


ao.o 


21.5 


1858 30 15.1 


6.9 18.7 


8. 


Sept. o 


22.6 


6. 


0.9 


as. 6 


25.2 


1859 9.8 as-8 


0.7 7-l 


19. 


Oct. o 


f|.0 


7-* 


4-8 


26.1 


0-9 


iMoB 


.7-6 


7-9 


11.6 9.1 


4- 


Nov. o 

Dec O 


0.9 

4 1 


7 

9t 


13 5 

111 


3 




47 

v 


186* 


3.6 2<>.2 15. a as. 9 


O.I 


LTV.V. V J'J 


* ' 


- t y i 


1863 10.4 | 10. a 


9.0 6.9 


11.3 


*ln January and February of Icap-yeara. 


1864 B 18.3 21.9 


37 16.3 


a 3 .6 


(he number* taken from Table II arc lo be 


1865 


s$.o 


3-0 


3-7 14.7 


7-6 


diminished by a unit. 


1866 


4-* 


13.6 


7-4 5-7 


18.8 








1867 


II. 


*4-3 


i.a 14.1 


a. 8 


TAHI.K 


III. 


1868 B 
1869 


18.8 
$.6 


6-4 
7-0 


12.1 3.5 
5-9 4-4 


15.1 
a6. 3 


Qerwdt of the Arguments. 


1870 


4.8 


7.6 


IS. 8 12.8 


10.4 












871 


11.6 


8.7 


9.6 si. 3 


21.6 




t 


D 


A 


B 





187* B 


9-4 


ao.4 


4.4 3-* 


6.6 














873 


26.2 


i.S 


14-3 * 


"7-9 


P . . 


a 7 -6 


9-5 


16.1 


7-4 


*7- 


1874 


5-4 


ia.1 


8.0 20.0 


1.9 


2 /' . . 


55-1 


59.1 


3* 3 


$49 


54 4 


1875 


11. 


22.7 


1.8 


I.O 


13.1 


3 r . . 


8a.7 


88.6 


48.4 


82.3 


ll.6 


1876 B 


0.0 


4-8 


ia.7 


10.4 


*5 4 


4 r . . 


no. a 


118.1 


64.6 


109.7 


108.8 


1877 


26.8 


IS- 5 


6-5 


18.8 


9-4 














1878 


6.0 


26.1 


o-3 


27. a 


0.7 








. 



















1880 B 


ao.6 


18.8 


S-o 


17.6 


16.9 








1881 


7-4 


9-4 


14-9 


6.0 


l.o 








1882 


6.6 


10.6 


8.6 


7-0 


12. a 








' 1883 


U 4 


ai.a 


2.4 


5-4 


3 4 








1884 B 


tt.a 


3-3 


13 3 


a4-8 


5 








1685 


0.5 


13-9 


7- 


5. 


9-7 








1886 


7-3 


84-6 


0.9 


14.2 


3-7 








1887 


M.O 


$-7 


10.8 


22.6 


15.0 








1888 B 


ai.8 


"7-3 


55 


4-6 


o.o 








1889 


28.6 


*7-9 


"5-4 


13.0 


u. a 









46 



TABLE IV. 

Secular Terms. 


TABLE V. 

Argument A. 


TABLE VI. 

Argument B (Empirical Term}. 


Year. n6z Diff. 


A 





B 





B fiv 


1848.0 


o.oo 





0.00 





o.oo 


40 . 4-0.39 




4- 0.20 












1849.0 


4- O.2O 


i 0.23 


i 


+ 0.34 


41 4- 0.05 


1850.0 


O.2O 
0.4O 


2 


- 0.44 


2 


0.66 


42 0.29 


1851.0 


0.6O 


0.20 


3 


- 0.57 


3 


0.95 


43 - o 62 






O.2O 


4 0.62 








1852.0 


0.80 






4 


1.19 


4-1 - 0.91 






O.20 


5 0.57 








1853.0 


I.OO 


4- O.2O 


6 - 0.44" 


5 


+ 1-37 


45 - 1.16 


1854.0 


4- 1. 2O 
0.20 


7 - 0.25 


6 


'47 


46 - 1.34 


1855.0 


I 4O 


8 0.02 


7 


1.50 


47 - 1.46 




4- o.io 










1856.0 


1.50 


g 4- 0.22 


8 


1-45 


48 - 1.50 




o.oo 










1857.0 


1.50 




10 0.42 


9 


1.32 


49 - 1.46 


1858.0 


o.oo 

1.50 


II 0.56 


10 


4- 1.13 


50 - 1-34 




o.oo 


12 O 62 








1859.0 


4- 1.5 




ii 


0.88 


51 - 1.16 






O.OO 


13 ! 0.58 








1860.0 


1.50 


O.OO 


14 4- 0.46 


12 


0.57 


52 - 0.91 


1 86I.O 


1.50 


O.OO 


15 


0.26 


13 


4- 0.25 


53 , - 0.62 


1862.0 


1.50 


O.O3 


16 4- 0.03 


14 


- 0. 10 


54 ' - 0.29 


1863.0 


1-47 


- 1. 12 


17 O.2O 


15 


- o-44 


55 : + 0.05 


1864.0 


4- 0.35 




18 i - 0.42 


16 


- 0.75 


56 0.39 


1865.0 


- 0.73 


- 1. 08 


19 0.56 


1 "* 


- 1.03 


57 0.71 


1866.0 


- 1.77 


- 1.04 


20 


0.62 


11 


- ".25 


58 0.99 


1867.0 


- 2.77 


- I.OO 


21 


- 0.59 


"9 


- 1.40 


59 1.22 






0.96 


22 


- 0.47 








1868.0 


- 3-73 


0.92 


23 


- 0.28 


20 


- 1.49 


60 4- 1.39 


1869.0 


- 4.65 


- 0.88 


24 - O.O5 


21 


- 1-49 


61 1.48 


1870.0 


- 5-53 


- 0.84 


25 4- O.lg 


22 


1.42 


62 i . 50 


1871 .0 


-6.37 


0.80 


26 0.40 


23 


- '.27 


('3 1-44 


1872.0 


- 7-'7 




27 


0.55 


24 


- 1. 06 


64 l . 30 


1873.0 


- 7-93 


o. 76 


2.8 0.62 


25 


- 0.79 


65 4- i. 08 


1874.0 


0.72 
- 8.65 


29 0.59 


26 


- 0.48 


66 0.83 


1875.0 


- 9-33 


0.08 


30 4- 0.49 


27 


- 0.15 


6? 0.53 


1876.0 


- 9-97 


- 0.64 
O.6o 


31 0.30 
32 4- 0.07 


28 


4- 0.19 


68 4- o. 19 


1877.0 


- 10.57 


n 56 


33 - 0.17 


29 


0.53 


69 0.15 


1878.0 


11.13 


0.52 


34 


- 0.38 


30 


4- 0.83 


70 - 0.48 


1879.0 


11.65 


- 0.48 


35 


- o.54 


3' 


1. 08 


71 - 0.79 


1880.0 


12.13 




36 


0.62 


32 


1.30 


72 - 1. 06 








37 


0.60 


33 


i .44 


77 I 27 




38 


- 0.49 


34 


1.50 


/ J ' * / 
74 - 1.42 




39 


- 0.31 


35 


4- 1.48 


75 - '-49 




40 


0.09 


36 


'39 


76 - 1.49 




41 


4- 0.15 










42 


0.38 


37 


1.22 


77 - 1.40 




43 0.54 


38 


0.99 


78 - 1.25 




44 0.61 


39 


0.71 


79 - 1.03 




45 0.60 


40 


4- 0.39 


- -75 




46 


4- 0.51 










*t w 

47 


0.33 






48 


4- o.io 




49 


- 0.14 




50 


0.36 



47 

TAIL. \ II, N**. 
Argu*ifHti } 1) ././ Ihf msnth. 








Mai. 


April. May. 


June* 


July. Aug. Sept 




-O.OI > 


-O.OJ 


-0.04 


l J -0.03 


-O.OI 


+ o 01 +0.03 


+ 0.04 +0.04 


+ O.OJ + 


1 


+O.OI +0.01 


+ 0.03 +0.0$ 


+0.08 +0.10 fo.ii +0.11 +0.10 


+ 0.07 + 0.0$ 


J 


0.09 o.io 


O. IJ 0.16 


O.I9 1 0.31 O.JO O.I9 


0.16 


0.13 o.io 


3 


0.19 o.n 


0.16 


0.39 


11.33 0.34 0.31 O.JS 




o.ll ' 0.18 


i 0.19 


0.33 o 38 


0.41 


0.49 ' 0.49 0.46 0.41 




0.39 


< *o.4$ 


+o.$o +o.$6 


+ 0.61 +0.66 


+0.67 1 +o 67 +0.61 +o.$6 




0.4$ 


6 0.63 


0.68 0.77 


0.83 0.8$ 


0.87 


0.8$ 


0.79 


0.71 




o.6j 0.60 


7 0.80 


0.87 I 0.96 


1.01 1.0$ 


1.01 


1. 01 




0.86 


0.80 


0-77 


8 0.93 


1.09 1.13 


1.18 ' i.jo 


II, II 1 


0.90 


0.88 0.89 


9 1.01 


1.12 I.JI 


1.37 ! 1.39 


1.37 1.31 


l.ll I.OI 


0.9$ 


o.ob 


10 +I.>4 


+ 1.1$ +1.1$ 


+ 1.30 j +1.30 


+ 1.36 -t-l.lS 


+ 1.08 +0.98 




il 4 0.97 


II 0.97 


1.07 I. It 


1.13 1.10 


1.1$ 1.07 ' 0.96 0.86 


0.81 


0.83 


13 0.81 


0.9! 1.03 


1.06 


I.OJ 


0.96 0.86 


0.76 




0.63 


o.frfi 0.73 


13 o.$8 


o.6( 0.77 


" ) 


0.66 o.$6 0.46 


0.37 


0.35 


0.48 


14 +0.38 


0.39 0.47 


0.48 I 0.41 


+0.33 


+ 0.11 


+0 11 


+0.04 


+0,03 


+ O.OQ 


+ 0.18 


i$ 




+0.08 +0.14 


+0.14 




0.03 


-O.IJ 


0.13 


O.JO 


-O.JO 


- 0.33 




16 


-0.34 


-0.3$ 0.30 


- 0.11 


-0.19 


-0.40 


o.$o 


-o.$9 


0.64 


0.61 


- o.$$ 


- 0.44 


17 


-0.60 


-o.$J -0.49 


-0.$J 


-0.61 


0.71 


0.81 


-0.89 


-0.01 


-0.89 


- 0.81 


- 0.70 


18 


o.8l 


-0.74 -0.71 


0.76 


-0.66 


-0.97 


-1.0$ -I.IJ 


-1.14 


1.00 


I.OO 


0.89 


19 


-0.93 


-0.88 -0.88 


0.89 I.OJ 


1.13 -1.31 


-1.16 


-1.13 


-1.11 


1.13 I.OI 


30 


-0.97 


0.9* 


-0.04 


i.oo 1 l.io 


1.31 1.17 


-I.JO 


-I.JO 


-1.14 


- 1.14 ' - I.OJ 


31 


-0.93 


-0.91 


-0.93, 


-0.09 


-1.0, 


-1.18 -1.14 


-1.1$ 


-I.IJ 


-1.17 


- 1.0* 


- 0.99 


33 


-0.83 


-0.83 


-0,86 


-0.91 


I.OI 


-1.09 -1.13 


-1.13 


I. 10 - 


- 0.9$ 


- 0.87 


33 -0.69 


0.70 


-0.71 


-0.78 


-0.87 


-0.94 


-0.96 


-0.0$ 


0.91 0.86 


-0.78 


0.67 


24 ~0.$l 


-0.$4 


-o.$9 


0.64 


-0.71 


-0.77 


0.79 0.77 


-o.7J 


-0.67 


0.60 


- 0.5$ 


35 -0.37 


-0.39 


-0.43 


0.48 


-0.$4 


-0.59 


o. $9 o. $6 


-0.$J 


-0.47 


- 04' 


- 0.37 


16 ] -0.13 


-0.17 


-0.31 


-0.36 


-0.40 


-0.41 


0.41 O.j8 0.34 


0.19 


0.1$ 


17 -0.14 


0.17 


o.jo 


0.34 


-0.17 


0.18 -0.16 -0.33 -o.io 


o. 16 


-O.IJ 


O.ll 


8 ' -0.07 


0. II 


o.ll 


1 -0.1$ 


o. 16 


o. 16 -0.14 


o.io 0.09 


0.06 


- 0.0$ 


- 0.0$ 


9 -0.03 


0.06 


0.07 


-0.08 


-0.08 


0.06 


-0.04 


O.OJ 


0.00 


+0.01 


o.oo 


jO 


o.oo 


O.OI 


O.OI 


O.OI 


o.oo 


+0.01 +0.04 


+o 06 +0.07 




+ 0.0$ ; + 0.01 


3 


+0.0$ 


+0.0$ 


+0.07 


+0.08 


+o. 10 


0.13 0.1$ 


0.16 0.14 


0.13 


O.ll 


0.07 


3 


+0.13 


+0.14 


+0.16 


+0.19 


+ 0.33 


+0.16 


+0.17 


+0.16 


+ 0.1J 


+ 0.30 


+ 0.17 


+ 0.14 



Nont. Each column is computed for Ihc middle of ihc month, but may be used for the cnlifc month without an 
error crer exceeding o."o. If much greater accuracy than this is required, a boritonial interpolation mut be used. 



48 
TABLE VIII, Sv. 

Horizontal Argument, or Argument at top, -Dg, or D "+30. Vertical Argument, g. 



g 


o 


I 


2 


3 


4 


5 


6 


7 


8 


9 


IO 


II 


12 


'3 


14 





































0.23 


0.30 


-0.36 


-0-39 


-0-39 


-0.35 


0.2S 


o. 19 


0. 11 


0.03 


+O.O2 


+0.03 


O.OO 


O.O6 


-0.15 


I 


-o.39 


0.46 


0.50 


-0.53 


0.50 


-0.44 


0.36 


0.28 


o. 19 


O.I2 


-O.Og 


-O.Og 


-0.14 


O.22 


-0.31 


2 


-0.54 


O.6o 


0.64 


0.62 


-0.58 


-0.52 


-0.43 


-0-34 


0.25 


O.2O 


o. 19 


0.21 


-0.27 


-0.35 


-o 45 


3 


-0.66 


-0.71 


0.72 


0.69 


0.64 


-0.56 


-0.47 


-0.37 


-0.31 


0.27 


0.26 


-0.31 


-O.J8 


-0.47 


-0.58 


4 


-0.75 


-0.77 


-0.77 


-0-74 


0.67 


-0.58 


0.48 


-0.39 


-0.34 


0.31 


-o.33 


0.38 


0.46 


-0.57 


0.67 


5 


-0.79 


0.81 


0.80 


-0.74 


-0.66 


-0.55 


0.46 


-0-39 


-0.34 


-0.34 


0.36 


-0.43 


-0.53 


0.64 


-o.74 


6 


0.80 


0.81 


0.76 


0.70 


0.60 


0.50 


0.42 


-0.35 


-0-33 


-0.33 


-0.38 


-0.47 


-0.57 


0.67 


0.76 


7 


-0.78 


0.76 


-0.71 


0.62 


0.52 


-0.43 


-0.34 


0.30 


0.28 


O.3I 


-0.38 


-0.47 


-0.57 


0.67 


-o.73 


8 


-0.71 


-0.6S 


0.61 


-0.51 


0. 2 


0.32 


-0.26 


O.22 


O.22 


O.27 


-0.35 


-0.45 


-0.54 


0.61 


-0.63 


9 


0.62 


0.56 


-0.47 


-0.38 


0.28 


O.2O 


-0.14 


O.I2 


-0.15 


O.2I 


0.30 


0.38 


-0.47 


-0.53 


-0.58 


10 


-0.49 


-0.41 


-0-33 


O.22 


0.13 


O.O6 


O.O2 


O.O2 


O.o6 


-0.13 


0.22 


0.30 


-0.37 


-0.44 


0.48 


II 


-0.33 


0.26 


o. 16 


O.O6 


+0.03 


+0.09 


4-O.II 


+O. IO 


+0.04 


0.03 


O.I2 


0.19 


0.28 


-0.33 


0.33 


12 


0.18 


0.08 


+O.O2 


+0. II 


0.19 


O.24 


0.25 


0.21 


0.15 


H-o.07 


0.01 


O.IO 


-0.17 


0.18 


0.18 


'3 


o.oo 


+O.IO 


0.19 


0.28 


o.35 


0.38 


o.37 


o.33 


0.26 


0.19 


+ 0.09 


o.oo 


-0.03 


0.04 


O.OI 


M 


-1-0.17 


0.26 


0.36 


0.45 


0.50 


0.51 


0.49 


0.41 


0.37 


0.27 


0.17 


+0.13 


+0. IO 


+O.II 


+0.14 


'5 


+0.32 


+0.42 


+0.52 


+0.59 


+0.63 


+ 0.62 


+0.59 


-1-0.53 


+ 0.43 


+0.34 


+ 0.28 


4-0.23 


+0.23 


+0.24 


4-0.29 


16 


0.46 


0.56 


0.65 


0.71 


0.73 


0.71 


0.66 


0.58 


0.48 


0.42 


0.35 


0.33 


0.33 


0.37 


0.43 


17 


' 0.58 


0.68 


0.75 


0.80 


0-79 


0.76 


0.68 


0.60 


0-53 


0.46 


0.42 


0.41 


0.43 


0.48 


0.56 


18 


0.67 


0.76 


0.82 


0.84 


0.82 


o.77 


0.69 


0.62 


0.54 


0.49 


0.46 


0.47 


0.50 


0-57 


0.65 


'9 


0-73 


0.80 


0.83 


0.84 


0.81 


0-75 


0.67 


0.59 


0-53 


0.48 


0.48 


0.49 


0.56 


0.63 


0.72 


20 


+0.74 


+0.79 


+O.SI 


+0.80 


+0.76 


+0.69 


+0.61 


+ 0.54 


4-0.49 


+ 0.47 


4-0.46 


+0.51 


+0.58 


+0.67 


+0.73 


21 


0.70 


0.75 


0.76 


+ 0.73 


0.67 


0.60 


0.53 


0.47 


0.43 


0.41 


0.44 


0.51 


0.58 


0.65 


0.71 


22 


0.64 


0.67 


0.67 


HO.62 


0.56 


0.48 


0.42 


0.37 


o.33 


0-35 


0.40 


o.47 


0.53 


0.60 


0.67 


23 


0.55 


0-57 


0.54 


+ 0.49 


0.42 


0.35 


0.29 


0.24 


0.24 


0.27 


0-33 


0.38 


0.48 


0-55 


0.60 


24 


0.45 


o.43 


0.39 


+ 0.33 


0.26 


0.20 


-1-0.13 


4-O. !2 


0.13 


0.17 


O.22 


.0.31 


0.40 


0.46 


0.50 


25 


+0.29 


+0.28 


+0.22 


+0.16 


+O.IO 


+ O.02 


O.OI 


O.O2 


+O.OI 


4-0.05 


+0.14 


4-0.24 


+0.31 


+0.36 


+ 0.37 


26 


+1.15 


4-O.II 


+0.06 


O.OI 


O.og 


O.I4 


o. 16 


o. 16 


-0.13 


0.04 


4-O.06 


0.13 


O.20 


0.23 


0.23 


27 


O.O2 


0.06 


O.I2 


O.20 


O.26 


O.30 


-0.31 


0.30 


0.22 


-0.13 


O.O4 


+0.04 


+0.08 


+ 0. 10 


4-0. 08 


28 


-0.17 


0.22 


O.3O 


-0.37 


0.42 


0.46 


0.46 


-o.39 


-0.31 


0.22 


-0.13 


0.07 


O.O2 


0.03 


0.07 


2J 


-0.31 


-0.38 


0.46 


-0.52 


-0.57 


-0.59 


-0.54 


-0.47 


-0.39 


0.29 


-0.22 


-0.15 


-0.13 


-0.15 


O.2O 


30 


-0.45 


O.52 


-0.60 


-0.66 


O.6g 


-0.66 


0.60 


-0.54 


-0.44 


-0.36 


O.27 


0.23 


0.23 


0.26 


0.32 



I'.t 

1 \ i \ I II --- Ciiiiliiin.-.l 

+ JO. / 






o 
I 


-o.*$ 


-o.$t 










O.3O 


-0.3 











-0$6 




, -o 71 


-o.6 






-0-4* 








-O.J6 




3 

t 


-o.6l 
-0.7*. 


-0.7* 


-0.78 
. -0.81 




-0.67 


o.ta 












S 


0.81 




o.$ 












6 


-0.80 









-o 47 




-0.6o 








7 


-0.77 










-0.42 








8 


-0.6, 














0.63 








9 

ii 


o 60 
-0.46 

-O.JO 


-0.41 
0.35 


-0.38 

















-O.IJ 




t 0. 12 




o.oo o.oq 












13 


+0.03 












Ofl 








'i 


0.30 


o.s 


















+0.37 


*-o.si 


4-O.5J 


















'7 


0.51 

0.63 




0.65 
0.7-1 




0.6, 










0.31 


o.3i 












0.78 






o.Tt 






















0.77 


0.80 


0.81 






o. jt 





















+0.77 


+0.80 






+ o 6ft 






1 
















0.76 


0.76 












o.t 


















0.70 


0.68 




0.55 




t^ 




0.30 












o.6 






0.60 


o. S 


0.51 








+0. It 
















*4 







o.tb 




















*S 


fO.JJ 


+o.7 


-t-o.lq 


+0 O-i 


O.OU 








-0.11 












36 
*7 


o.lS 
4-0.01 


0.061 


+0.01 
-0.lt 


-0.08 


1 




-o.M 










-0.14 


-0.3* 






~o.$o 










9 
3 


-o.3> 
0.41 


-0.4* 


-06., 


. 


-o. S 8 


. 
. 



7 M 



50 



TABLE IX. 

Argument, g. factor to be 
multiplied by n Sz. 


TABLE X. 

Argument, u. Factors for correction of latitude and reduc- 
tion to ecliptic longitude. 


'_ 


F 


u (F-l) 


WJ 





o 


+ 0.118 


o 


- O.OO4 


+ o.ogo 


o.oo 


i 


0.114 


i 


- O.OO4 


0.088 


0.03 


2 


0.103 


2 


- O.OO3 


0.081 


0.07 


3 


0.086 


3 


O.OOI 


0.069 


O. IO 


4 


0.065 


4 


+ O.OOI 


0.054 


0.12 


5 


+ 0.040 


5 


+ 0.003 


+ 0.036 


0.14 


6 


+ 0.015 


6 


0.004 


+ 0.017 


- 0.15 


7 


0.009 


7 


0.004 


0.004 


- 0.15 


8 


- 0.034 


8 


0.004 


- 0.024 


- 0.14 


9 


- 0.054 


9 


+ O.OO2 


- 0.044 


- 0.13 


10 


0.072 


10 


O.OOO 


0.060 


0.11 


1 1 


0.086 


ii 


O.OOI 


- 0.074 


0.08 


12 


0.096 


12 


0.003 


- 0.084 


0.05 


13 


O.IOI 


13 


- 0.004 


0.089 


O.O2 


M 


- 0.103 


'4 


0.004 


0.089 


+ 0.01 


15 


- 0.099 


15 


0.003 


0.085 


+ 0.05 


16 


0.092 


16 


O.OO2 


0.076 


0.08 


17 


0.080 


17 


O.OOO 


0.064 


0.11 


18 


0.065 


18 


+ O.OO2 


- 0.047 


0.13 


19 


0.046 


'9 


O.OO3 


0.028 


0.14 


20 - 0.024 


20 


+ O.OO4 


0.008 


+ 0.15 


21 


+ O.OOI 


21 


O.O04 


+ O.OI2 


0.15 


22 


0.026 


22 


0.003 


0.032 


0.14 


23 


0.051 


23 


+ O.OOI 


0.050 


0.12 


24 


0.075 


24 


O.OOO 


0.066 


O.IO 


25 


4- 0.094 


25 


O.OO2 


+ 0.078 


+ 0.07 


26 


o. 109 


26 


- O.OO4 


0.086 


0.04 


27 


0.116 


27 


- O.OO4 


o.ogo 


+ O.OI 


28 


0.117 


28 


O.OO4 


0.089 


0.03 


29 


O.IIO 


29 


0.003 


0.082 


0.06 


30 


+ 0.096 


30 


O.OOI 


+ 0.072 


- 0.09 



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4- .386- 
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- .401 
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4-.39S- 
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180 
7S 
170 


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165 
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- 359 
314 
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4- .351- 
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- .116 


- -*35 


4- .111 - 


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.021 .055 


.091 


- .188 


- -'93 


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.105 + 


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75 
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