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^m
A TRANSFOmrATION
OF
HANSEN'S LUNAR THEORY
COMI'ARKn -VSITIl TIIK
THEORY OF DELAUNAY.
SOrON NEWCOMH,
SUrKUI.NTKNDKNT AMKKICA.N i:i'lli:.\|];|{l.S,
AIDKK HY
JOHN MEIER,
ASSISTANT AMKlilCAX El'lIEMKIUS.
57
TIlAXSFOIlMATloV uF fl.WSKNS LINAK THEORY,
■a
I
TIk- imni.Ticnl .•u:ni.utation ..r' rlu- iiuMiunlirios in tli,. uHMm's , notion (.x.-ruK-d bv
II.VNSKX was prolKil.ly tIk- ^TeaK-st st.-p taken in ivrmt tin,,... tuuanl pl.-nin^ tin- tl,.'-
()iy ot tlie Immr i.ei-nirhatioi.s o» nn ac-iiiat.- nnnicrical l.asis. It uas tlu- sf.-j, nlii.-h
first reiuU.iv.l it r.-rlaiu liiat .my .Hsrivpancv betuecMi th.- tlM-..ntical an.l ul.st-nxl
values ot tlic ni..,,ualiti.-s pr...hi.-..-.l hv rlu- Min an.s,- in.m s.nnc utl,,,- caiiM. i],m). trr-.r^
111 tli...ry. 'I1,e tlie-.i-.tical valii.^. f, nhicl, it k-tl nnist !..• .•unsidMrd tlu- nn.st a.vii-
rato wiiicli astmnonn- ij<»\v jj« .».s.>>t<e>.
Tlio only tlKM.iy wliich .-an f,,niiu-r<- uitl. IIanskn's is tliat of I )i:i,ai nav. H.^ro
tlie coodicicuts aiv .K-v<.],.j.<,.l In ,,,,-{.■> .-onv.T-in- s.. sl,,v,lv that sonu- of tlu- r.-s.ilts
are still ., little .l.Mil.tfnl. iiotwithstuinlin-- the -ivat .-xtent to which the apj.n.x-
inuition was eamo,!. It may U, expeet-.l thai the numerieal theory on whi.-h Sir
(JKOKcn; AiKV is now eu-ji-v-l uill form y^ an.,tluT step in advann-, in' which iiotldnjr
will Ije wanting- lor tlu- imijH.>t-s .,f accurate astruuoniy, so that three tlu-oiies of tliT-
hi-hest order (pf .-, -curacy will uhimately l,e availal.K- for the eonstniction of l„uar
tables. The work in .ple^rk.iJ UAw^r >rill uiitinishetl, the residts of jJAXsh.v aiul Dklai;-
NAY are the oidy ones now avaiiaMr^
rnfortnnately. tlu- tlie-TV .,f FFanskn cannot I.e directly compared with those
whi(diliave precc-ded it. owin^r to fli^- peculiar form oi' the variables in .vhich the
co-ordinates of the m<H.n an- .-.xpres.,-d. In sayin--- this. 1 do not contest tlu- proj.o-
sition tliat this form h;is a<ivaiita:r».-.-i. l!ut, apart from the cpn-stion of its merits in
forin, it becomi-s important U> have the means of making- a direct comparison of Han-
sen's theory with that of hi> |.re.leees.sors ami c(daborers, who have e.\presse<l the
co-ordinates of the nu.oii .iirecfly in terms of tlu- tinu-. This has twice lu-en partiallv
done: by the writer in the 0.«»/./*. li^mlii^ for i86S, I (T(.me LX\"I, p. 1197,, ••""h
indepeiul(-ntly. by Schjki.lkkj-i.. itt a paper published in 1874 ])y the Danish A( ideniv
of Sciem-es. jioth dej.eiul on data for the transformation oiven by IIansk.n' 1 imself,
Avluch, thouizh tlu-y may b,,- ac<-iinite enou-li to -,nve an idea of the aj^reement between
the^ theories of IIanskn and 1»el4I-xay. camu.t be re^-arded as sulliciently precise lor a
satisfactory transformed theory. The object of the present paper is to^make a ti-an>-
formation which shall faithfjilly represent IIaxsen's latest theory, aiul be exju-essed in
Jirgumenls dopending- directly on the time.
59
6o
TKANSl'ORMATION OF HANSENS LVSWs. THEORY.
I'm
v^ I.
i:\l'l!i:SSI()N OK TIIK MOON'S lAJNGITI UK.
In IIax-skn's tliLM.ry tlir iiuu.ii's louj-itude is re|»n.-^-i.j..-.l in tliu followlnj-' form.
I/, tli'.» luooii's moini jmniiKily ;
I/' . tilt' Sim's lui'iiii iiiitiiiiily :
c), tlic (listiuicc troiii the uniU- to tlif ]i('ri;:«'-' :
(,}'. tlic ili>\..i('.(! from tile ikmIc I(. the x'lar jM-ri'^iee;
T, the lon;4'ituil(' of tlic ] n tI ;:■(>(■ :
r. tlu; iH'ceutrii'ity of liic iiio'>iiV (H-liit. as ux-d l.y Hax-SKN ;
)iS.:, the Iliiiiscuiaii ptM-tiirliatioiis of mean aiK-maly:
.s'. tlie n.insciiiaii |icrtiirliatioiii> of liuitiKle:
I. tlie iiicliuatioii of the moon's orhit.
'I'hen |)Ut, as iiuxih'ary (|uantitu's,
./ — elta (',// + i"'^.-), tlie true anomaly
tan
., 1
I sin 2 (./ + o.>} + ' tail' I ^iu 4 /-^ «> — etc
fhe retlnctioii to the eeli[)tie;
,,, tan 1 eos (,/'+ m)
I — snr 1 snr ( / + m)
— o" 397 ^i" 2 CO
— I ".198 sin (2//'+ 2 co')
— o".2S5 sin ( 2 ,'/ — 4 //' 4- 2 &J — 4 cii'X
the iiiei|Ualities of this reihictioii.
Then, for the moon's loiigitiido,
K-/ + ;x+ K-r I!-
The latitude, fi, is <;iven by the equation
1
1
sin /:/nsin I sin (/+ w) 4- -*.
In presenting' Hansen's results in the form <if a. complete and exact numerical
theory, several precautions have to lie taken, hi the first plate, all the results must,
so fur as possible, depend upon or be reduced to one and the same hcniogeiieous set
i
TK.VNSI-ORMATION OF HANSEN'S LUNAR THEORY
6i
of elcnionts. Tn tlir next pliuv, tlu.s,. iiM..,unIiti..s wl.id, ..xpn-ss tl„, sulutiun ui tin-
prohlci.i ot tlii-ce iMMlios, nnisi.lcre.l as iniitcrial points, must 1... s.^arated fn.ni iin-
quahtu-s arisinj.' tVun. utlicr sources, sucli, tor instance, as tlie .listan.'c l,..tn-eMi ti.e
moon's centres offrmvity an<l fio-ure. and tiie eilipticity cf tli<. eartli.
Tliree values of tin; eccentricity appear in IIanskn's tlie(.rv and tables:
(1) A provisional or ideal <-ccentricity, with which fhe ine(Hialities w-re ori-n-
nally computed. "
(2) An apparent eccentricity, uliicli he found to represent the ohserv.-.l motion of
the moon's centre of (iuure, and used in his tables.
(3) A theoretical eccentricity of the true orbit described by the moon's c( ntre of
f^ravity.
Tiiese three values of the element are: —
(i) r — .05490079
(2) r = . 05490807
(3) '==.05489959
_ Accordinjr to IIanskn's view it is the third value which should be used in c,nn-
puthifT the moon's perturbations; ])ut as he actually used the tirst value, it is tia- on-
which we should employ in tlie transformation.
In the case of the inclination there are three correspondino' values, with an addi-
tional complication arisin..- from the (pu'stion whether we shall add to the inclination
Ji term in the perturbations, 2'.7o5 sin (//+ a,), havinj.- the mean arf^unient of laTitU(h-
<is its argument.
Omitting this term, the values of the inclination will be :—
(0 1 = 5^ 8' 48"
(2) 1-5' 8' 43".66
(3) I = 5" 8' 39".96
Here, again, it is only the hrst value with which we are con< ■ ; od iii the tran>fonna-
tion, because it is the one employed by IIanshn in coin[)uting '.; perturl)ations.
The IlanseiiiiUi perturbation >t6~ is ai xplicit function of 7, //', a) and co'. So
lar as the longitutle is concerned, our present [)robleni is to express./, R and R', and
thence L, as explicit ♦unctions of the above four quantities. If we put: —
2 =. f/ -}- uSz
Ci, f.,, ?:„ etc., the coefficients of sin „-, sin 2^-, etc. in the development
of elta {e, z), we shall have,
f=.^-{-Ci sin ^ -f (',, sin 2^ + etc.
62
TRANSFORMATION OF HANSEN'S LUNAR THEORY.
If, thon, WO put // + mU tor ^, (lovclop in |M)\vei-s (»f «r5,r, call {'\ (f),, tlu* part of /'iii-
(lopcndnit of i/<'>:\ and (c, ,7)^ tlio (•(.(■niciciit id' {u<'^::)' in./; wo sliiiU have,
(«', r/)„ rr // + '\ sin // + r, sin 2// -f- r, sin 3// + r, sin 4// + etc.
(/>,//), = 1 + r, cos // + 2(:, cos 2// + .V', cos 3// + etc.
1 . 2" . 3-'
(c, y).. = — ,, ''i f*ni // — ^ r, sin 2// — ^ r, sni 3// — etc.
I 2' 3'
(f, //);, = - ^ , ( , cos // - - (■, cos 2// - ^ T;, cos 3// - ftc.
-" »> *3 *• * J
(f,//), n-—'', sin // + etc.
etc. etc.
The coeilicicnts r,, r,,, etc , lU'c (Impendent o)i tli(! eccentricity. Tliu well-known ana-
lytical \aliies. and tlu' ninnerica! values obtained hy outtin;-' c rr .05490070, are:
^., - 2 c - ' c'' + ^^ r'' = .10976024 = 2 2639",676
''•= = 4 "' ~ 24 '■' + {h ''" - -^^-^^ -'^''-^^^ = 7 7^^"- 269
64
I 2
10
.> ,/l
96 480
'097^5
960
— .00017893 iz: 36". 907
=: .00000972 =: 2". 005
— .0000005 7 = c". I 1 S
\
900
— .00000004 ■=. o .007
Tlie value of ;m5^ is taken, not from IIanskn's tables, but from liis revised results
{riven in the Ihirlff/uiiii*. They are found in I'art I, pp. 409-411, and I'art 11, pj).
224, 242, 25S, and 26S, and, for convenience of reference, are all collected in Table
1 of the present [)aper. In this talde are •i'iven alsd the powers ot' lu').:, the coinputa-
tions of which were ail made in duplicate, that of the scjuare bein;^' executed by t
inde))endent comj)uter.>
wo
We thus have all the data for the numerical value of f, the formula for which \>
Consider next the tirst term of K, which we niay call Ki. We have
(0
K, = — tan- 1 sin (2/+ 2 w),
which is also to bo developed in ])owors of Hf5,r.
^ I'lMlcr lliin Ullf n I'lTciiii' is iniiilc \i> Hiiiiscn'n Iwo papers, Itiirleijunijihr lliioirlixcheu liirichnutiij tUr in dm Miindta-
filn toKji'iviiiKlUK Sliintiiijni, in the Ablmiidlunijni dir kiiniijlirh-mchsischen OeuUmUuft dcr ll'inKiiixdiafkii. Hand /A", Xl.
TUANSI'OR.MATION OF HAN.SI:NS LI'NAR TIIKORY.
63
Tf WO .sul>Htitut. lur/ ifs v..,l,u. in h-nns of . ..hI ., .,,.1 dovel..,, in ....vc-rs uC .
If, = — tiin- I X
24
+ .^c
:\
12
■sin (—2.; ■\-2(.i)
sin (- ^- _|. 2 o)
+
0-
(^=-s'0
'c-'+ „r'' I sill 2 0)
4
■^' +16''
j sin ( c + 2 «)
j sin ( 2Z -\-2c0)
+
+ (^4 '■"- 24 '■ J'^'"( 4-4-2^0
+ J^^^' sin( 5-- + -''^)
+ ,6 '^ •'^i" ( 6c + 2 a))
Tf, in this o(jniition, we snl.stitute for r and \ tli.-ir numorical valnos and tl.on
ddlorentiate witl, respoct to ,:, so as to ol.tain tl.. .•o.-frn^icnts of tl.o i^owers of h6:
luttmy
wc have
1m = 1{, „ + 17, , ^/A',: -f Ifi , (;/,"),--)-' -f otc,
'm.o = — o".oo6 sin {— fj -\- 2Go)
— o".942 sin ( 2 &»)
+ 45".62 7sin( // + 2 oj)
— 41 1".626 sin ( 2 /y-|- 2 &))
— 45"-2''^i sin ( 3// + 2 ta)
— 4". 040 sin ( 4 ^y -)- 2 Qj)
— o".33S sin ( 5 // -f 20))
— o".02 7 sin ( 6 /y + 2 &>)
K, 1 = + 000 221,2 cos ( /y-f2 0j)
— .003 991,2 cos ( 2 // -f- 2 f.))
— .000 658,6 cos ( 3 // + 2 &j)
— .000 o;8,3 cos ( 4// + 2 w)
— .000 008,2 cos ( 5 ,*/ + 2 G))
— .000 000, S cos ( 6// + 2 &))
• 'TiiWfsof (his and tlio otlior ilcvcldiiiiiciils in lli.. (■Ili],ti.' niiilioii li,i,. ,m.-« ;riv.-n liy t'rof.-.s<.r Cayi.ky in flio
M,:m,ni:'< of Ihe Ittnjal Anlroiiumical XooWi/. \,.l. XXIX, l.nt il„. „buvc dLM'l..imiL>nt was .•xe.ntcl in.I.'|.rn.l.-ntly hefort)
till' iiitplicability of I'lolVusor Cavi.kv's loriMiilir wa.s ivniarkcil.
64
\v<
TRANSFORMATION OF HANSEN'S Ll-'NAR TIIKOUY.
U, ... = — .0001 I Kill ( // 4- 2 co)
-f- .00399 sill ( 2 // 4- 2 f.i)
+ .00099 m\ ( 3 // 4- 2 a))
4- .000 1 6 sin ( 4 ^ + 2 cj)
li, :, = 4- .002 7 COS ( 2 // 4- 2 &')
-f- .0010 COS ( 3 /y 4- 2 <»)
In the same way, jjuttinj,'
U..:=itan'i- I sin (4./ 4-4'«')
liuvc by siiltstitiiliii';' Inr,/' its viiliu; in ,;, and (U'VC'l(ii»iny in powers olr,
i
sill (4/4- 4 ^''
1 1
('■ sill (2 c 4- 4 m)
Puttinii' as betbro,
— 4r sin (3,~4-4<y)
4- ( I — 1 6 e") sin (4 ~^ 4" 4 <")
+ 4 r sin (52 + 400)
4- ^- e" sin (6 ,^ 4- 4 go)
11, - 1:,,, 4- II, /M5.: 4- 1{,,, (;/(5.-y- 4- otc.
we iiml b\- snbstitiitino- the numerical vahies of I and r
K.J,, = 4- o".oo7 sill {2 // 4- 4 t.:))
— o".092 sin (3 // 4- 4 &))
4- o".400 sin (4 // 4- 4 '«^)
4- o".092 sin (5 // 4- 4 r«')
4- o".oi3 sin (6// 4- 4 ^)
— .000 001 ,3 cos ( 3 ^r/ 4- 4 '«')
4- .000 007,8 cos (4 // 4- 4 ^')
4- .000 002,2 cos (5 // 4- 4 Gii)
The terms of li.o (»'^c)- are less than o".ooi.
The coeflicient uf — s tan 1 in li' is, Avitli suilicient accnracy,
cos (/4- ai) [14- siir I sin" (/4- «))]
or
( I 4- sin" I J cos ( f-\- 00) sin" I cos (3/4- 3 ^)-
V 4 / 4
1
1
I
TRXNSI'OKMAiioN or HANSKNS \J\\U rilKony,
\W til.' (Icv.i.|.i,„.iits,,ft|„. ,.|li|„i,. „„„;,,„ ^,,. 1,.,^,.^
C.S ( y-f ,o) = — '^ r-' CMS (- 2 ,• -f ai)
s
' '" (
•(IS ( — ,: -j- m)
-{■ ( \ — r- ) cos ( ;■ -f- ^))
4- ('■— ■\''') ens (2,; + <o)
. 9 .,
+ ^, '■• <'os (3 : 4- a))
+ ^■' cos (4: + f.>)
COS (;,,/■+ ] a<) —
21
(•- + 3"')
— ;, '• (■( IS (2-4- 3 <.))
+ (1 — (jr-) CIS f ; . . ;, M)
4- 3(' CIS (4:4- • c)
If WO repi.'S(..iit liy S tli(! (•..(■Ilici.-iii nf v ii, IJ , that is,
S — — tan 1 cos (./'4- ^') ; 1 4- sin- 1 sin- (./■4- a,)) J,
S = S„4-S, ^,r5,:4-S,(/.r5,;)-,
and suppose
we shall liave,
65
S„ — 4- .0000 :;4 cos (— f/ 4- r.))
4" -^0495 5 ^'"^ ^'
— .0899 78 cos ( // 4- U))
— .004936 COS (2// 4- (O)
— .000306 COS (3 // 4- &))
— .000020 COS (4// 4- G))
— .000030 COS (2 // 4- 3 G))
4- .000 1 ;6 COS (3^-1- 3 t^)
4- -000030 COS (4 (/ 4- 3 ft))
4- .000004 *■'"* ( 5 // + 3 &')
S, = 4" -0900 sill ( // 4- (,})
4- .0099 sill (2 // 4- &>)
4" .0009 sill (37 4- co)
S, — 4- .045 cos ( // 4- 00)
4- .010 cos { 2 // 4- f«')
Mjiltiplying- these several exiiressions by I1an.skn'.s s, we find the value of .s' 8,„ etc.,
given in Table 11.
2
#^:i-'*v.-
wV**^-**':
55 TRANSFORMATION Ol' llANSICN'S LINAR TIir.OKV.
Collecting all tlio cooHirionts of the i...\vers ..f ^mV;, wc lind the following expres-
sions for the moon's true eeliptic lon-itii.le, as a function of /m^ ; :—
Terms Unliti mlrni n/nS:.
-}- 22639". 676 sin //
-f 776". 269 sin 21/
4. ;/j".907 sill 3//
4- 2".oo5 sill 4//
-f o".i 18 sin 5//
+ o".oo7 sill ()(/
— o".oo6 sin (— // + 2 co)
(— o".C)J,2) .
: • sill 2 f.)
(- o".397^
-f- 45". 62 7 sill ( //+ 2 f«0
— 41 I ".626 sill ( 2 v -|- 2 c.))
— 45".2Si sill (3,'/+ 2 ^))
— 4".o1o sill (4// + 2 f«>)
_ o".33.S sin (5//+ 2 a>)
— o".o27 sin (6//+ 2 (.))
+ o".oo7 sin ( 2// + 4 r*))
— o".092 sill ( 3 // + 4 (->)
-f- o".400 sill (4// + 4 ^))
+ o".092 sin (5// + 4 <-«')
4- o".oi3 sin (6// + 4 &))
— l".iQS sill ( 2//'+ 2 (o')
— o".285 sin t^// — 4,'/ H"
f«)
4w)
('(icffic'u lit (if II '5,:.
(The ic. 111111:1 pilillls llIT ~i\ llhlrrs iirdrciiiiiils. I
-)- .109700,2 COS //
4- .007526.9 COS 2 11
4" .000536,8 cos ^n
4- .000038,9 cos 4//
-■f- .000002,8 cos 5//
4- .000221,2 cos ( // 4- - ^•')
— .003991,2 cos (2// 4- 2 f«>)
— .000658,6 c(»s (3// 4" - f'^)
— .000078,3 cos (4// 4- 2 M)
~ .000008,2 COS (5// 4- 2 r,))
— .000000,8 COS (6// -[- 2 <i))
TRAN'SroRMATION OF HANSEN'S LUNAR TlllCORV. 6?
— .oooooi,;, CDS (3// -f 4 (.))
+ .000007,8 (MIS (47 -f 4 G))
+ .000002,2 ('(ts (5^ + 4r,))
+ - .^,
I--.. =: — .054SS sill f/
— .0075;, sill 2//
— .oooSq sill 3 f/
— .ooooS sill 47
— .0001 I sii. ( // -\- 2 &))
+ ■00390 sill (2// -f 2 («))
+ .00099 sill (3// -\- 2 (<■>)
+ .00016 sin (47 -(- 2 Mj
I..;, — — .01 8,^ COS //
— .0050 COS 2 /■/
— .0008 (.'OS 3 //
+ .0027 l'(»S (2// -f - &>)
+ .0010 COS (3// -(- 2 &))
'II1C several parts of this expio-^ioii tor L are i^iveu in Table II, oniittiiin- rlie fo]-
lowiii;;- terms, wliicli are, however, all inchuled in the coliinin i^'iviiiL"- the concluded
coellicieiits in L : —
1. The terms ot' L,,, explicitly liiveii in the tirst ol' the precedinn- c(jnations.
2. The exiiressions for //'V;, { 11 ''):)- X "^ ^„ (//'') .:)-'X 'm,:;- fi"d ii/'')-) X \<:.u
The values ot' die last three expressions are as t'ollows. the nuniliers within the
parentheses heiii;;' coetlicients of//, 7, m, and cv/, respecti\(d\- : —
» '5 „- X IL I
{,iS,:Yx^^2
{»'y:fX 1^,:
— .001
sill
(3,
1
01
\
— .002
sin (0,
^
— 2)
— .00 1
sin
(2,
I
"1
0)
+ .001
(',
2
2
2 )
+ ■0'^3
sin [2,
O
— 1
2
-\ \
+ .001
sin
{2,
— I,
- 1
0)
— -005
(-^
2
">
2 )
+ .002
sin (3,
O
— 1
-^
2 )
— .002
sin
( *-',
o
0,
-^)
— .020
(3.
1
2 )
+ .002
sill ( —
I, 2
c
2 )
— .004
sin
('.
n
0,
2)
— .005
(4.
->
T
> )
+ .(304
sin (c.
'>
-»
0,
^
— .002
sin
(2,
n
0,
2)
— 002
(.-
I,
4,
0)
— .002
sin (2,
_ |,
4,
-^ )
-f .002
sin
(2,
2
4,
-2)
+ .002
(4,-
I,
4.
0)
— .002
sin (3.
^
4.
— 2 )
+ .003
sin
(3,
— 2,
4,
--)
+ .001
(.i, -
I,
4.
0)
— .002
sin (4,
-^
'>,
— 6)
+ -003
sin
<4,
•^
4,
-:^)
— .003
(4. -
'1
6,
- 2)
-• .002
sin (,■',
-<j,
C),
— 6)
•j- .002
sin
(5,
4i
--^)
+ .016
sin
(5, -
T
6.
-2)
— .002
sin (2,
-6,
4.
— 6)
— .001
sin
(1.
— 6,
4.
-6)
+ •013
sin
(0, -
6,
- 2)
— .002
>i"i (3.
- 6,
4,
— 6)
— .001
sin
(2,
— 6,
4.
-6)
+ .002
sin
()•,-
T
6,
-2)
— .001
— .001
sin
sin
(6.
(7,
— 6,
— (^,
-6)
-6)
68
TRANSFORMATION OF IIANSKNS LUNAR THEORY.
Til T.-ililc II tlic (•(tliiiiiu "Sum'" contains tlic sums of tlio torms actuallv iriven in
the pivccdinn' coluunis of the talilc
TIk' uc\t column ^ivcs the com]ilct(' cocllicicnt of cacli term in the ecliptic lon^i-
tmle. and is t'ormed l»y addiiiu' to the column "Smu" the omitted ti-i'ms just referred 1o.
The last c(dunui j^ives. for the larger terms, the elements which the\' principally
contain as factors. If these; elements he chaniied, the coellicients nuist he chiiniied by
corresponding;' (plan titles.
vS 2.
IJKDrCTIOX OF THF, IMM'.CIvDrNd KXIM.'KSSIOXS TO ('XFFOini KLK.MKNTS, AND
COMI'AHISON WITH I>KI.ArXAV.
The coeHicients of the precediui>- incfpndities contain as factors certain elements
for which different investigators adopt ditl'ereiit values. It is essential to a (dear pre-
sentation of results that they should l)e reduceil to a uniform and W(dl-define(l set of
elements havinii- j^iveii v;diH-s We tlierefoi-<' coinmeiu'e hv i-educlnu' tin; theorie.s of
both Han.skx and Dki.ai NAY to such a system. The (dements principally referred to
are
mam
(a) The ratio of the mean motions of the sini and moon,
(fj) The lunar eccentricit\'.
{)') 'Hie solar paralla.K.
('M The solar eccentricit\-.
(f) The iiudination of the moon's orbit.
Iveally, all tlu'se (dements are coiitaiiieil in all the ineepnilities in a very complex
ler. Ihit there is so little doul)t about their true numerical vahu's that it is only
necessary to take account of their (dian.u'es uheii they appear as factors in coefHcients
of coiisideral)le man'uitnde. The extent to whiidi ea(di tei'm is affected can l)e nai^'hly
seen from its analytic e\pressi(m -iveii by 1 )ki..-.i:nav at the end of hi.s n,orir du
Moitniurnl <lr la Liit/r, 'i'onie II. We take up the several elements in oi'dei'.
{a) Untiu „f iiH'ui/ iiHiliniis. This (dement is so certain that no reduction need be
made on ;U'counr of it. It is true that theoretical mof'ioiis of the lunar n(!(le and
peri<4('e imist implicitly enter in coniKction with this (dement. Mut. fn.m a row^h
examination (,f I Iansf.n's integration coeliicicnts on pp. 350-352 of his /fmirf/iirif/'^ J
do n..t flmik any (d" the laruer coetlicieiits will be aOected by as nuudi as ' of
»i • • , , . " 100000
tiien- ennre amount by any adunssible (diaiinc uf these motions.
(/;') Errndnvitii of mu,„>\ orhif. The eccentricities used by the tw., investijrators
are not directly comparable, but may be nio.st convenientlvVompared l)y nHhicing
each to the coetlicient of .7 in the expressi.m for the moon's' ecliptic lonnjtude. J)i.>
i.Ar.VAV uses Aiuv's value, niven in his la.st paper (.n the (dements of the nmon's orbit*
Il.VNsi N corrected his eccentri.dty for u.se in his tables, as alrei.dv mentioned. Tl
writer (^tained a sniall but w(dl-marked c.,rr(.ction to IIan.skn'.s yuiue from the Green
• .Nli'inoirs lioviil AHtioiKiiiiical Socioty, Vol. X.XI.X.
le
TRANSIORMATIO.N OF ll.WSKXS LUNAR TIIF.DRV. 69
wicli ol)sorvi:tioiis 1846-74, niid tlic AVasliiii;,Hoii Dbscrvatioiis iS62-'74. Tlio four
values of tl"0 coi'lliciciit in (|ncstioii arc: —
AiiiV, used hy Dki.ai NAY, . _ . . 22639". 06
Hanskn, used ill 'riicor\-, .... 22637".! ^5
I Ianskx, used in TjiIiIcs, .... 22640".! ^
CoiTcctcd \alnc found in 1876,* . . 226:;o".v8
Altliouj^-h tlicrc is 110 ivasonaMc doiiht that llic ('('ccntricitx- ..f 1I\nsi-:n's tables i-e(|uires
a iieg-ative coi-i-cction, it will Ix' adopted for the ]air])oscs of coinpai'ison Ijceause it is
tu)\v the standai'(l ot the cplicniei-idcs with which ^niis(M|UciU coinpai'isons unist Ik;
made. All the tei-ius ha\ iui;' c as a eoellieieiit, nnisi thci'cfoi'e hr' ineivased Itv tli(^
factor
.00000728
zr .0001 ^20,
.05490
and those liaviii;^' r'-' l)y d(Uil)lc tins I'actor. The coctlicients in r nnist, in 1)i:i.ai;nav',s
theory, bo increased hy the factor
I .09
12639"
.000048:
{y) Solid- juira/ld.r. II a\si;n\s theory does not set out with a defiiute solar iiarallax,
but with a ratio of the in<'aii distanci's of the sun and ukmiii, which ratio ayain is not
the usual one, because IIanskn's h and n' aic the same functions ot' the motion of mean
anomaly that the i'sual '/ and n' ar(-(iftlu' sidei'cal mntinns. W'c must tl.ei'efoi'e adopt
an indirect pro: ess foi' iindiiiL;- thr i'clatii)ii n\ solar pai'allax and paralhn fie eijuation on
his theory. He finds that his thcoi'ctical coctlicimt has to be nniltiidied t)\' the t'actoi'
1.03573 b) make it aLi're(» with obseiAation: and then, in ^^ 260 of his l)(ir/c(/Hi/(/, he
deduces the solai' parallax 8 '.(1150. DividiuL;' this paiallax by the pn.'cedinj^' factor,
we coindude that the [larallax ol his theoi'\- is: —
8". 6085.
In turnini^' his tlieorv into numbers I)ki,\i".\ay used 8". 75. The parallax to which
both theorii's will be actualK i-eiliii'cd is: —
8". 848.
Hence, TIansi-.n'.s tei'nis hasiuL!' the pai'allax as a factoi- inust be inci'eased by tho
factor
.02785,
and Di'.f.AiNAv'.s by the factor
.on 20.
('^) 7'lie solar <■( cri/tr/iifi/^ The solar eecenti'icitx' ot' 1Iansi;n\s tlieorv is: —
c' =: 0.01679226 (I'^poch !8oo).
J'ii|ii'r.-i piililislicil li,\ till' CiiniiiiiK.siiui or; llic 'liuiisil, (if \'i'mi,s. I'uit III.
ro
TRANSFORMATION OK IIANSKN'S LUNAR Tlll'OKY.
!)i;LArNAv uses la; \'ki{Kiku'.s value :-
c' := 0.0167 7 1 06
(Kpdcli 1S50).
Ill strictness tlicsc twn values are not coniiiarable. owini;' to the diilerenl form of
Hanskn's solar tlieorv: l)iit since IIanskx ne;^lects |)erturl)ations of the earth's motion
in his liniar theor\-, it ina\ he assnnie(l that there Mill he no (lill'erence between the
form in which the eccentricity enters into the two tlieoi-ies. If we carry Lio \'i:iiHii;K"s
eccentricit\' hack to iSoo with his secular \ariatioii, we shall ha\e : —
0.01679228
(K[)(K'li iSoo).
This iiia\- he re^anlod as a!»olutely identical with IIanskn'.s value for tin* same e{)ocli.
So, ailoptiiii;' I Soo as the epoch, we lia\e oiilv to iiurease Dki.ainav's coeilicients in
<■' hv the hictor
.00002122
"707677^
.001 205.
Or, we ma\- re(lnc(; Hansen's values to iS^o In- dividiiii'- them hv 1.00126s, when
0)
lie\' will lie comiiaraole wi
th I)
KI.Ai'.N.W S.
lie riieories o
f IIanskx and hKLAiNAV, thus reduced to a uniform and consistent
set of elements, are <iiven and compai-ed in Tahle III. I )i;lai:n'Ay'.s results are fre-
(juentlv doul)tt'iil h\- a small fraction of a second, owin;^- to the slow converi^eiice of
the seri<'S in powers of m, and the table has been arraiij^ed so as to show the extent of
lie iiiicerramtv tliiis ans^ui;'.
l'"ollowinLi' the indices expressing;' the arguments are LiiNen. lirst, Hanskn'.s coeiH-
cients formed from the \aliies in T
lie 1 1 b\' iniiltij)lyiiin' ii\-
lie a
ppidpriate I'actors tor
reduction alreadv ui\eii. Thev are onl\- iiiven to o".oi, but should the thousandth of
se
ccnids lie reqiiiivd they are readiK' obtainaljle.
'I'll
correspoiidiiiii' coeflicieiits of Dklai.'NAV are (lei-i\ed principalK' from his |
ire-
.sentation of niimerical results in the additions to the (
niiUtitssinicc I
On
11 to 21 of that
paper are j^'iveii the si
/c.s T('i;iji.s tor 1869.
iinis of the terms in each coelHcient
which were actually computeil by him. The parallactic terms, as^'iven bv Delainay,
are s
till t
o lie multiiilied b\'
1 - ,\
i + A'
A beiiiir the ratio of the mass of the moon to that of
the earili. rutlinu', with Hansen, A — , the coellicient
80
rected for this coeflicient and for diiferenco of element
will he
8i'
Tl
le sums, cor-
th
s, are f^wvw m the column
fli
I>('/((Hi/ni/ (\). Had all the appreciable terni.s been actually computed, tlie.se coetli
cieiits would have been the delinitive ones of Dei.aunay's theory, lint it was fre-
([iiently tbiiiid that the terms, even of the ninth order, where the development c(!ased,
were still appreciable: it was, therefore, necessary to estimate the ))robable sum of the
0111
itted terms of lii"lier orders from the law of tl
K! series as (deserved in the terms
actually computed. These estimates can have no true niatheinatical fouudtitiou,
TUANSI-ORMA riO.N OF IIANSENS LUNAR TIIKUKV.
71
1)Ocanso tlicre is 110 proof of tlio iictiiiil law of the scries.* .Still, tlioro is a liii)'li doj^Toe
of probability in favor of each one iicinn' ■•'■ '•'"^'^ " ''"'l*' ajipi'oximatiou to the truth.
A rigorous coniputation would probal)ly show that a majority dilfcred less than ' of
tlieir amount from tlie true \alues, though here and there one mij^'ht be found entirely
illusory. The coellicicnts of l(in<iitudc, nioditied by these estimated additions, arc
f>'ivcn !)}• 1 )Ei,ArNAV on paj^cs 38-40 of the paper rct'erred to, and are reproiluced, with
the necessary corrections for chaiin'cs of elements, in the colunm Dc/aiUKi// (2).
The ditference of these results, ^iven in the next colunm, is the correction
ai)])arently applied l»y Dklacxav for tlie uncomputed terms. It will bi- noted that we
liave no indei)endent statement of these tei'uis to refer to, ami can onl\' infer their
values fnun the ditfereiu'cs lietween the printed I'esults (1 ) and (2)
Finally, we have the difference, I/ai/^n/ minus DihuDitiii (2) >liowin^- th(^ dis-
crepancies still outstanding;- between the two theories l'',ach one can jndi^e foi' himself
liow far these discre[)ancies arise fi'om the uncertainty of Dklainay'.s senn'-empirical
corrections, and how far tVom errors in the two theories.
. One or two terms are woi'thv of a special examination, and amonii' tii(>e the par-
allactic C(piation takes the liist I'ank, a> upon it depends the value of the solnr [)ai'allax
to l)e derived from a ^^iven observed value of this ecpiation Arrani;'iiij.;' 1 )i:1jAII.nav\s
terms according- to the power of ///, which enters as a factor, the result will be that
<4'iveii b(dow under the head I',. Dklau.nav omits terms in y- after ///', ami t(.'rms in r
after in''. Corri'ctinii' the result i'or an estimated valm- of tliesi' terms, derived b\' in-
duction, we shall have those i;i\-en umler the heail !'_.. It will be seen that the terms
follow a nearly reyular law up to in'\ but that hi' deviates from this law. Assumin<>'
this term to be in error, and estimating' the valr ot' it and the higher terms as those of
a, ycomcitrical proi^-ression with the ratio we have tln' residts 1';.
Po
p.
Terms In //^ — 73". 1760 — 73". 18 — 73". iS
iir
Sum
\ -o^J-i — 34 -o
o
I 2 .01
34 -30
12 .01
III' — 12 .0082
w/' — 4 .()8i2 — 4 .50 — 4 .50
///"' — I .9815 — 1 .89 — I .89
111^' — o .7122 — o .72 — O ."JZ
— o .48
III' — o .381 I
o .72
o .3S
127 .242-
I 26".c)S — I 2 7".oS
Our choice nnist lie between the results P. and 1';,. ff we ado])t the fornuir we may
add o".26 as an estimate of onnttiu"- terms <,^iving' : —
1'
r=-l27".24; P'=:J^P=:-i^4".
TO.
• W tiKiy lie iviMiirli.'d I1i;il in llir sni.'s U>v \\\i- .-niiliir ;iici'I,t;iI ion Ori.u \\1 I'l I (Iw Iniiis ol' ii lii^liii- (Udcf
ac'liiiill.v 111 cliMMHi' llii'ii- -^iK"! (liivt ll.v ciiiiliiir.v m ll ^liiii.ili' xvliiili wmilil li;ivr luiii tMiiiir,! rmiii \\\<>s,- of ;i liiwrr
(iidiT,
TRANSFORMATION OV HANSEN'S I.INAR TIIKORY.
If we iulopi tlic latter we li;iv(!
P rz - i2;".o8 ; 1" = :,; 1' zz - I 23'. 94.
Si
Miiltiiilyiii;^- l»y the eoellicieiit i.oiu to reduce to the i)ariilliix 8".S4S the result will
be:— '
(-^) -i-^5"49
(3) - i^5"-33-
IIankkn's coetHcieut, — i25".43, I'mIIs hi-tween these; results and may he regarded as
certainly corn.'ct within less than o".i.
The other term referred to is that ilependin<i' on the ar;^-ument: —
of which the principal parts of the coellicienr are, in Dklai'NAv's th(M)ry, —
3(1 order,
4th order,
5th order,
6tli order,
7th order,
8th order,
9th order,
Do
.10
- 5"-^o
+ 10'. 1 5
+ 9"-34
+ 5"-^2
+ 2".90
+ i"43
Dhi.ai'nay seems to have taken i".i8 as the ])r<ibal)le sum of the omitted terms,
whereas they should have been taken as o".94 to agree with Hanskn.
S^ 3-
LATITLTDK.
Taking Hansen's expression lor the moon's latitude :-
sin /y = sin I sin (/ + gj) -f s ;
the first step is to form the expression sin i f -{- m) in terms of //, ro, etc. This may be
done in two ways. By the first we express the required (piantity as a function of ^, and
TK ANMOkMATION < >K IIANSKNS 1.1 N.\K TllloKV.
tlK'H put <l I- /mS; r,„- ,/ ,.,,„1 .!..v,.|,,p j„ p.nvcis ,,(' i,<S:. I'.v tllr lll..,„y
lit' cllijitic
sill ( /'-I- r..
Oj;
— , r sill I — > .; -[■ Ml
9210 ■ .^ • 1 /
>lli (- 4,; + M)
-^(~,:s''--k'o"';)-'^---^-)
-( - ,,'^ + l^ '■'') ^i" (- 2.: -fa;)
r sill r.)
+ (3X4' 19^ '■")'^'" ^5- + -)
Si
40
r' .sin {6 .: + (>'')
+ ,- ,, '- sill 7,r + m).
«
Tl" we now siil)-.tltiiii_' fVtr .-. »/ -|- >"*> :. t">r r its niiiiu'rical \;ilni', and
[mltiiii;' : —
i|i-\ fluli.
we sliall liave
^in I .-in 11'+ (r = V --p F, ,><^ : + F_, ( i, <y:)- + F, (n 'y:)''
F„ ( ill arc = — o' o I 2 sin ( — 3 // + .'.>)
— o -255 sin (— 2 // -(- r>))
— 6' .96S sill ( — (/ -f m)
— ioi5".S34 sill f.)
-f eS447".342 .sin ( // + r>))
-f- roi2".oii sin ( 2 // -j- 6,j)
+ 62 .45S sill ( 3// + r.i)
-f- 4 .061 sill ( 4 /y + <i>)
74
TKANSI'ORMATION t)!-' HANSEN'S M/N \R THEORV
(ill Jirc) (CoiitM) — +
+
+
o".2J2 sit) (
o".oig sill (
o' .001 ^•ill (
F,, (ill radiusi — — .(.")0(> oooi sin (— i a -r ot)
— .000 00 1 2 sin ( — 2 ,7 -f «•>>
— .000 o^^3S sill (— 'J-^*^)
— .004 9249 >iii '''
+ .o.S9^352 sill (
-|- .004 90<j4 sill (
-|- .000 ;i02S >iii (
-j- .000 0197 >iii I
-f .000 0013 >iii (
i7 + «>
3, ff + 'y)
4 <•/ -f <y>
A> a dice
adoptcMl,
Then
-}-. 000 OOOI sill (6//-]-")
F, — + .000 0002 COS (— 3 fi -f <w)
4- .000 0025 cos (— 2 7 -f **'>
+ .000 03 3 S CI tS [— ff-T f^}
+ .089 4352 cos( .7 + *^)
+ .009 S I 27 COS ; 2 '/ + <")
+ .000 90S4CI.S1 3 5r4-i»>
4- .000 07 S8 cos ( 4 .*/ -r a>>
4- .000 0066 (MIS ( 5 .'/ -f ^)
4- .000 0005 cos ( 6 ft -r <")
!•'._, — 4- .(H/O 02 sin ( — // -r &-'»
— .044 yz sin ( '/-+- '*•'''
— .009 S I sill ( 2 // -4- <w>
— .001 3O .-in ( 3 V 4- oxj
— .000 16 sin ( 4 '/ — «^*
— .000 02 sin ( 5 </ ~ oi)
F;, iz: — .0149 cos ( 7 4- <y)
— .0065 cos ( 2 ,'/ 4- <^'*
— .0014 cos ( 3 ,7 4- OJJ
k upon the value of sin i sin (./'4- wj a >ec<.]i(i uieth*>fl ot" coniinitin;^- it was
as toHous. li"t lis put : —
<yi'—/—'i.
sin (,/'4- f'A ■= sin f ,7 4- ro 4- 6/}
— cos 'S /'sin (v + '*ij
-f sill ("i./cos {</ 4- <^'j-
TRANsroKM Aiiox ,,|- || sssKNs I.l'NAR Tllli.KY
75
Fn.M. tl... ,,..,„..,•!,..] vain., of <\r .Wr.Ay .iv,.,, tl.- p.uMS uf tliis .,,nntitv w.r.
tonued, aiul tluMicc its cusun- and sine tn.n, the luimnla. :—
cos 'W = I — ■ 4- (>ti.
1.2
sin <S / ~ ,S / ~ ■' J.. ,,!,._
1.2.3
'J'licsp ..xpn-ssions wciv fln-n nnilti|.li.Ml l,y t\w sine and cosinr ..|' (,/ + m).
'I"l'<' incan dilV.ivncc Ix'tuccn the cM-iHcicnts in sin I sin (,/ + r..) Innnd l.v fli..
two nictlK.ds was l(.s> than (i.e.,;,. the lar^vst one hitin^' ".oiu.
Addiii-' IIanskn's s In this cxpicssi,.!! we havi- the vahu- <.!' sin //. 'I'hfu A itself
is (ihtaincil l.y the inrniiila
A zzrsiii /y-f ' sin' /5' + ■'' sin" /i.
The pniicipal |.arts,,|' /; aiv o-ivcn In 'I'ahl,- |\\.,f which the rulunnis .L-t'emnj^- to
IIanskn's tliciirx sccni lu nccil no cKijIauatiun.
vs 4.
iMJtrcTioN (»i-Tiii; i.ATin Mr; ani> comcaimson with kiilai nay.
.Ml the terms i\\' the latitude CMiitain the inelinatinn ol' the niDon's oi'hit as a factor,
and are therefore to he niulti|ilied liy such a constant coellicient that tin' |)i-inci|)al term
ot the latitnde shall aLiree with ol)>er\ atioii. 'The transt'oi'ined expressions of IIanskx,
L:i\-en in Talile i\', leail to a consistent theory in which the otdficient of the |ii'iiicipal
tenuof the latitmle is iS4oV'.24.S. The expressions ,,f Dki.ai xav also lead to a theor\-,
in which tliis coellicient is iSpx'.jo. i'lacli of these is to i)e multiplied li\- sncii a
iactoi- as shall reihice it to the \alue iniplicitl\- adopted in I Ianskn's tables. There
JIansk.n adopts : —
1 = 5" S' 39". q6,
which is less hy S'.oi than that of the theorw Hence, from this alone wonld tVdIow
the correction : —
— S".o4 sin ( ,/'+ '«')•
Hut, the tables contain, anioiiL;' the pei'Inrliatioiis, two terms which depend maiidy on
the same ariiinneiit, namely • —
2". 705 sin (./"J- a>),
wliicli, developed by putting;' // + 2 c sin // for./i appears as a perturbation, and
3'''.7o sin (// + f.)).
„^^ linNSFOiniM-InN nl IINNSKNS I.i;N.\K lliroKV.
l.li..lM..nnl.u,.l,o,l,.. ..i.u.ni n,Hs.,r(l.u,v.u,l..r,n.i,v..r,l,.
„„„„, Tl.. >Hin of llu. Iir>t two ..xpivsMnn. l-m.- ^Irvlu,,,.!. l..rnHK.
_ 5".;, ID >ill *,'/ + ''')
-fo".29.:; sill M
— o".:g2 sill (J '/ + f«'.).
Ad.liii- tlir lliinl, 111'' I'nu ill//-}-'.' will l'."'"iiic
- l".6l(; sill I// + '.'^.
W,. .iv iw. ron..,.nH.,l .ill: iIh' -n.- in - .H'l ^ '/ f '.-■ 'Hi. ;:iv;il..r |,;.rt -f ih^ir
,,,,,,,,,,,,,,,,,,,. ..,.n>i,l,.,va as ;i r'- l-'-tnH-.-H. .ln.M,Ml,Mi..iiv ul U^
in.plirillv .•..il.i.H.l ill 111. Ml,l.. Inil iiu, h^lunuin,' U, ih. pn.M.m ut tliivr InMirs.
Willi the liist cinvctiuii tlu' t.-nn in '/ |- '•' Immm.uics
iS.jOi ■.'):o sill (.7 + '«').
.vliicli i- tli.-.'H.'llinMil iiiipiinlK cMiPiiiMMl ill ll\N>r.N'sl;il.lrs. ,,,.,.,
T„,lii> ,li,. writ.rtniMHl :. mnvrti..,, .4-. ,'.15 iVuni ( Mv-iiwi.-l, ;ni.l ^\ i.slini^'ton
..h.Tvntioiis 1SM.-7.,. iMit it will 1„. ,v,;,i,n.i witliuut riiniiuv. II.mht ;i11 llir rnHh-
ri..|it> in 11ansi,n-> /A fis -ivn in T;ihl.' IW mv to 1h. iliniinis!,..! l.\ tlic tiirtor
.ooooSS.
iiud thoM' nf l>. l.M-NAV .IIV tn lie inrrc;l<C(l li\ tllf liu'tcr
.000020.
•n„. „.nns In , .•nnl -' ...v to !„■ iiiodilicd liy tlic snnic .■orniciMits .s in ill.- .-isr ..ftlic
l,,„,_,iln.l.'. TIm Iv h.iMn> wliicli will I- iiltrcrlMlilv uWrvwA liv tli.' cluin-v ..fr rm'
tlioM' (l(M)fii(lin:.;' nii <.> .■iiid : .7 -j- '■'.
'I'Im. 1 liii.'atiniis hen- iiKlicat.Ml liav not hccu niadr in tin- ivsnits. l.c-aiisc tli-y
iuv SI. >ii-lit, and alVi-cl su lew tiTiii^. tliat carii ,iiic ran make iImmii lor liiiiiM'lt.
I'lic ciluiini Ihhunniii ( 1 j (■..mains, as l.ctorc. tli.' mum ..I tli- l.aiiis a.tnally .•<.iii-
niitfd \)\ hi-.LAi-.sAV. am! -iv.-ii liy liiin in tin' Cuiiiniissiuins ilr-< I'viiij^s l.ir iSog.
In Volmiin Ihhniinni ; : i In- fli.a.'iit.- arc .•..iMvrt;'.! l.y the lii-li.'r Utiio, of wlncli
the value lias liccii .•stimal.Ml l.y iinlii. •lion, 1> K iN.v liinix-lf .li.l ii..t -i\«- tlu'Sf
iul.litiuiis,*!;.) that they liad to he cstiinatcl '.y llii' writer.
vN 5.
I'.\|;AI.I,AX.
IIwskn's tli.'orv Li'ivcs till- pi'i'liirliatioii^ of ilic natural jo-'ai'lllnn of tli.' 111. ton's
railiiis v.M'tor. wlii.'Ji aiv tin- n.'Li'ativ.- ..f tli.' pcrtnvl.atioiis of tli.' l..,^'antliiii sine
narallax. The valiif ..f ir. in sccnn.l.- ..I' arc, i> foiiinl in tlic l)iir/r</ini'i, I'art 1, pa^cs
409-41 1, ami i'art 11. paii'fs 224-226, 25S. ami 2OS. 'riic in.it.ii's parallax /i i> .u-i\cii
1)\- IIanskn uiul.T til.- form
. I) (, .,|-,. cos/)
l..n- sni y, -!<.-■ ^, (^j_^,:^ — '".
11'! ANSI oUM.\l|(i\ oi' ||\Ns|:.Si's I.iNAi; l|||;.»RY,
n
'" "''''■'' " I- tll<' I'iMlillS nl' tllC ..artll lit IIm' lltilinl,. nf u|,i,.|, (llr >ll„. is s,f\.A\xA>l
','"■ '"■> Hl.-.ll, .liMMIlrr il: l!„. 1Ln>,,m,, ll.ru.v. SvMrl, i-, ,|;ii;.,Vlll n, .h.|llliliull
''■ ''"• ""•"" 'liMMiHT n{ tlir (,r,|ii,,.,rv il ri-s. " li is not. Ii.,wvcr. iHTf^-.irs' t..
'''■''""■'■ ''"■ """ '" '' 'Ii"i' (l!i'-lly. Immmh-.. tlirv iiMv In. iiM.>I si,tUr;ict..nIv n.iii-
l""''''l '•> 'I"' ^"l"'- "I' til'' '•n:i>t;,n| mC |,,m|';iI|;|\ Iu w'j ilrli 'l i|,.\ l.^ni.
< 'liiiii.Uiiio' the loL:;irItliin> t,. luiiiii.ij (jiLiniino :iii<| >lfvci(.|iiii-- in |M.\\ri-- (.f ir, tin-
;il)ii\(' i'.\|)i't's.si()ii ;^i\cs: —
>\\\ II
l'~
I ) \ -l- r ('(IS /
r' I
r cos/ / ^,.- V
;iiiil tlii'ii
, SIM'/) ,
III (Ic\clu|)il|u' , !•,,> / i^V(( lii(ll|ip(|s of cniiilillt.-llinli wi fc llxd. iis ill lllr c i||| j ill t;i t inll
"I' til'' |irillci|i;il Icrill 111' llic liititiplr.
I. I'Vdlll ( '\\ I.I \'> IllMl-, \\c ||;|\,.
(•(.s,/ —
-1 .:■■
cos 4 ,:
, ''^^5 I
+ r ens
:v^4
;iii(l then l)y siil)>titiitin^' // f- // '*> : i\>v : \vc Ii;i\c ens /■«lc\rln|)cil In iiiHl;i|i|cs .,1" //, etc
2. I'llllill"-
\vt' Iiavf
/~-.'/ + 'V
ciis/'zz ciis '"^y <'<>"i// — !^iii '\/ sill //.
I)
I lie \ nine (il w ;i- (1( ri\til li\- 1 1 a\S;-,n iVoin tlic Icii;.;tli ut' tlic >ccip]iil> iiciidnlnni
(I ■ ■ '
Mini the (linicnsiiiiis ut' tlir ciirili iis iuiniil li\' IIksskl. The ili-i'ix.itinn i.■^ l:!^ i-ii in tiic
A^lruiKiiii'isclif Stichrirhliii, \ ulunii' X \ II. |);il;(.' .VJ<-'. 'i'lic <l;ila in. hie use nl' nn-: —
I), rjidiiis nl'cjirfli inidi r llic |i;i' illd aw sin s^' \ . . 6370063 nicti'cs
I*, Iciin-Hi ol'si'CdiHl^ ]i'Mi(liilnni miller s;iinc jiiii'.illcl . O'''. 992666
I
hi, mass 1)1 1I1C iiicinn •
78
Tliti rrsiili is
TRANSFORM AlIiiN Dl ll\Ns|:NS I.INAR IIIF.OUY,
I>
liiy zz .'■'.J I 71 II ,U>.
!Ir j;iv('s ii.-- the rfsiiliiii:^' (•(nisliint pari nl liic >:iic ut tlic |iariillii\
iiml tlic cliaiiL.'!'^ Ill fit'' ('oiistaiit |)ruiliircil li\ ->iiiall cliaiiL^'o in tin' data
Iiicrcax' (it 11""". I III I' \ai-if> tlic cdii'^taiit li\
liicrrasc lit' loiK)'" in |> \ai'ics the (■iui>taiit Ii\'
liirrcasc dl' iiiiitN' ill <lciiiiiiiiiiattir III //'
rile ili'Vi'luiiinciit >iii)^rniiciitl\ 'nvi'ii jrails to a ruii<taiit i>t"
— O'.I I
4-0". 1 8
+ o".i7
3422 .09,
a result (i".(); ;.''i'catfi' than tiial >taIiMl li\- IIaN'skN.
1 11 (•iiiii|iariiiL: iIm' iiaraliaxo nl IIanskn am! I >i;i,.m \ai tlii- milv clcnirnt wiiicli
will iiiatii'iall\- all'iTt the i'r>iill i^ tin' (■Mii>taiit ul |iaralla\: a ciiniiiai'isiiii nf tlio
(liricrciit \aliii's ni' flii> nut-taut, wliicli lia\r liccii rrci'iilU ohtaiiiril, will tlirrcrDrc be;
of intcri's:. Tlirti' ili>tiiirt inclliods of ohtaiiiiiiLi' this impoi'tant clfinciit liaxc iircii
apjilicil
((»■). 'I'hi' thiiirclical iiiftlioil roiimlitl mi Kki'I.kk's third law lis cxpi'cssud in tlif
thi'orN 111' L:ra\ itatioii, and dt-rixi'd rniidaniriitalK Ironi tlir iMpiatioii
II II
-' = m + >r,
(I iiriiii;' the iiicaii distaiirc ol the niooii. uliirli is iiiiincdiatid \- coninTtcd with the
parallax: // thr iiuaii motion, oj' iIh- \aliii' of wliirli tin re is 110 doiilit, and /// and .M
the inassis nl'thc moon and earth, exiires.-i'd in ainu'oiirial
pnale units, the ileteiinination ol
whiidi is the nio>t doiilitliil part of the iiroldeii
(//). .Measures ol the moon's po-ition made at two dislant stations, and reduced
to a eoiiimoii moment.
(r). Meridian derlinalioii> of the iiioon maile al the same station, and reduced on
the li\potliesis thai the midisliirlied L!-eoeentric oi'liit is a i^reat cinde.
The last met hod is not w ell adapted to i;i\e a certain result, owiii;^- to the constant
errors with whiidi inuasiires ol' alisohite declinations are all'ected. ^\'e shall tlierct'oro
conliiie our consideration to the lirst two.
'I'wo deterininatioiis l»y method (i-ri. that ol' IIanskn, just (pmted, and that ol'
AiiAMs in the Mniitlilji Xnltcrs, \'ol XI 1 1, and the llritish Siiittkiil A/hkuhic I'or 1S56,
aiv a\ailaiile.
Id.
The data \\><cd li\- Mr. Ada
M.i are
I >, t'l'iim liKssEi., and iherelore the same as IIa.Nsk,\.
1',* ;,j-6 So 1
(J IMiLillsh teet, or
///, mass III moon,
o"'.992 7i :
81,
piililisluil jiaiiir
riiis \:iliu' ill Kiiglisli led wiis Kiiiillvo.iiiiiiiniiiHlcil hy Mr. Adamj, liiniMir, not bcingoxiplicilly nuotuil in IiIn
TK.\NSI„„M.\ri..\ ..!■ I, vNs,.N ,s , ,:n.\U TIII.oRY
with IIa.n.skn u.. |,,s„:_ .I^<IM„,,.^.: ._,,,. | „ .•un,p;,n.
79
|iiiif It
Clmii-.-iii |)-o, - - - cliin,-.. uf .T„ _- o
" " l' = -fo .040. •• .. _,/,o5
11 II '
= I- '.5,
•( n
+ o' ,26
^ •?' — — iy •'- - II->.:n-s .,;„„. ,1,1. n..,l, ,1 ,.„.m...r
;inn,MV ;"";,'^^,-"'";:";':'''7^- "•'•-•'••r- I^Nsi.:Ns..n,i,nl. M,n„ ,„■ ,i,:„ .1.-
'•';•'• '•'•;'""■ ""• -'l";> '""1: ..,• il .SH.:V nu.l A,.AMS ,0 II vvsKN's .1.,,. nl
iii^i to til., system aliv,i,|\ .•H|u|,t,.,|, i|„. ,vs„lt, ui|| !,,.;_
Cnll^tiiiit (.iMiir p;irillla.\, 1 1 AN>|.;.S. ■^^2-".nq.
" " 11 »
•^I'AMX. ;i4^2 .13.
('nllstailt nl' |.,irall,i\ i|,|.lt'. IIanskv. ;,.J-^2".2v
" Aham-. vp:? .2S.
'I'iK-cn.istaiit orn.(|„rtlni, jmni ,]„. si,„. ,,. ,|„. iMiallax iisi.|f is 4-,, i^-
//. TIlc llio.st ITrcIlt ilctcnililiatii.lis nf
llH' III, .nils jiarallas l»\ iiicasiirciiici
• 1 ,■ MI, M . , |'Mi,,ii,,\ n\ lllcasii|',.ill(|i| ;|i',>
''•7' "' •^''•- '""•■'•^- '■^'" ''^ l- A. S. X.XXIl, „nl „f .Mr. >p,.>..m!Ih,| \\\IV)
'""'' •"•'■ t'"iii.l,.,| HI, ( '..ij,,. nliM-natioiis ;,,„! |„,tl. lend t.. a .•.■ii>iai,l of
It ,s not ilistnirtiy statnl wlu.tlH.r this i> tl,.. ..unsfaiit nf ,|„. parallax Itself n, of i^
sine Ml'. P.UKKN's iiitrndurtinn ,1. ... |,,,, ,p, ,,7.s,.,.,i.. to iniplv that 1,0 u.rA Mr
Adams s expn'ssion l,,r sn,.. |,,.,,allax as tl,.. parallax itsril i„ n-iliicini,' thr Cape nl.sn-
vati.ms. Ilut.ii, t|„. ivihirti tl,.' (iivMiuirl, Ml,>MNatiM„>. In a pplics A l>AM,s's ,,,1-
rcctioii t(. the parallax ,.f Ann's lii„ai' ,v,li„tiM,,.. u|ii,.|, nu,.. il,,. p;in,llax itself
To put the matte,- into a.iuihe,' shape: ( )„ p. , ,,, )\y, D.^kkn has 342j".33 us the con-
stant of parallax. On p. i^j he h;,s a c.i.st;,,,, ,,,ri,.etioii of o".r,S to the Aiuv-I'lana
parallax, of whirl, the constant is 34.M .So. uhirh j-ives 3422 .4.S as the constant of
parallax.
We shall probably make a near appi'.,xiniaiio„ to the truth l,v a.-^sni,,!,,-. that .Mr.
r.KKKN'.H mean provisional co,,.stant was 3422 .40, a,,.l a.s h,,. (Ieiii,ce,| a co'nvciion ,,f
+ f"' .38 tliLs would j4i\-e IIS his result : —
('oi,sta,,t of parallax, .
Constant of si,ie,
- 3422".78
3 42 2". 62
Mr. Stonk also linds a correction of + o".3S to .Mr. Ada.ms's i)arallax. This
would !^i\e : —
Cons[ai,t ol" paralhux, .
Constat, t of sii,e, ...
- . 342 2".86
. . 3422". -o
The ovidcnco is thcrefoi'e ii, fasorof a positive eoi'i-ection to IIanskn's constant-
hut, in accordance with the practice in other parts of this paper, the results as printed
are all founded on Uanskn's fundamental data.
^
'J^
So
TRANSKdRMA I li)\ oF IIAS'SKN'S IJ NAR rill.DRV.
Ill rllc Tillilc \' till' ciilllinllS (•(ilil.lill-
) 1
(i). The viiliic (if • - - .,.( I + '' C's,/'), cxprfsscd in sccoiiils of arc.
)r.
(2). Till' i»!'(Hli!cr (if tills (|iiaiitity Ky — /'' + ,,
(3). Tile (■(icliiciciils fdi' IIansi-.n's sine parallax, fdniicil Iiy addiiiL;' ' l) ninl (2).
If the |iai'alla\ itself is rci|iiirc(l, it may lie fdiiiid li\- addiii,:; flic rcdiictidii fniin
tile siiu- to tiie parallax iiscll', iiaiiiciy : —
-|-o".i5; +o".oJ5 cos //
-f- o".004 CDS '// — 2//' A- 2 10 — 2 m')
4- o".oo4 cos ( 2 // — 2 //' + J <•' 2 m').
(4). 'I'iic cdclliciciits (if ! »i:i.Ar\Av's sine parallax. >o far .as actually coiiijuitcd by
liini. A> lie stopped at the tci'iiis of tlu' fifth order, tlic liiiii(lre(ltlis of scchhhIs are not
always dcfuntiN-c.
(5). The >aiue, with tile addition of (piaiitiiies e>tiiiiatei| hy iiidiictioii to reprc
sent the omitted t(.'niis of liiLiher orders.
(6) The correctiniis applied in the precediiiL;' colmmi to olitaiii the most pi'olia-
hle \ allies of the co<d'licients.
(7) 'I'he deviation of' IIansi.n's ciwtliciei!t< iVoiii the second set ol' 1 >i;lai'nav's.
As some of 1 ))■,!. AlN'\\'s teiiii.^ ai'" doidifl'ill tVom the iii-ufli'Mclif coiivei'^.i'eiice of
his series, the coeilicieiits of' .Voam-'s parall.ax. foimd in the Mmithlii Xcticcs II. A. >S'.,
\'ol. XIII, p. 2<)'-,. ha\c iieeii ad.ded for conrpari--on. It uill he seen that rhe\- at^'reu
(doselv with the coeflicieiits of IIanskx. thoiiuh deri\cd independently of llieiii.
rRANsKORMATiux oi- ,,.^^s,:^■s r.rxAR TiiKoKv.
8i
4' i-
II, \z
sin.
CdS.
sill |.()S,
sill.
COS. sin.
-3 -
+
+
<). I or)
7. "35
7-73S
I
-3
o
-3
I
-3
+
+
+
+■ (i.oii _ U.O05
+ 11.075 — o.i.i,^
*" 0.044 — 0.007
o.oir _
o. 240
O. T2()
fl.0I2
0.077
o.S4f)
o.-l.'^
o. 122
o.oofi
0.003
0.0^;
o . 00 1
0.014
0.003
0.002
. 002
0.003
— t
+ 0.003
— O.OOI
—
o.O(j4
+ 0.043
— 0.010
I
—
2.524
-{- O.OS2
- 0.03:;
2
—
0.052
4- <uo
- 0.024
3
— 0.002
— o.(,o4
— I
— I
-
0.040
,
+ 0.001
-'
+
3.665
4- 2.324
4- 0.046
I
— 1
—
27.620
+ 15-144
+ 0.0S5
2
— I
—
23.00')
r S.4()9
4- 0.045
3
— I
—
'.337
i- o.(;S4
f- 0.007
4
— I
—
0.060
-r 0.04I
-2
— 2
—
0.020
-- 0.003
^'
-2
-
i.S()3
- 0.0S5
^ 0.007
-2
—
41.64^
+ "-3'.);
4- 0, 762
'
— 2
-i-
4466. icjj
4- I.S67
+ 2.3.SI
2
— 2
+
2i44,,),j5
■f- I 4')l
t- 1.750
3
— 2
+
''10, 021 »
•f- r,.420
-r 0. |l J
4
1
+
2.0S3
- 0.071
t- . 1 S
5
— 2
•f
0.0S4
+ 0.00 I
6
— 2
+
0.004
•
•
-I
-3
—
0.0S2
— 0.006
-3
-
2,3;:'
- '.437
+ 0.061
'
-3
■f
li)S. 103
- '4.S"
*- 0.216
2
-3
-r
155. "47
- 1) , 1 40
4- 0.202
3
- 1
+
5.1 6(1
- 1.341
+ 0.0511
4
-■ 3
4-
0. I(;S
- 0.047
4- 0.003
5
-3
+-
. 009
— 0.00 I
•
O. I 1.)
+ o
(- o
— 3
— o
— O
.012
.604
■W)
.1 = 2
. OOs
. I()2
, .,3s
.1)05
, "22
"75
(,06
11)5
064
043
-f-
— 0.004
— 0.017
-t- 0.010
— o.of.j 4
— (J.3)(i -r
— 0.036 -
-t- 0.002
-f- 0.0 ',s -i-
— '•77'J -
4- o.iif, _
— 0.005
4- 0.00')
4- 0.27)
f- 0.05S —
4- 0.003
— 0,00a
. 4-
0.002
0.004
0.001
(.1.003
O..J2I
0.071
0.022
o . r)02
0.004
O 007
o.t)02
0.003
(./,U)'
COf.
2
I
2
-\-
0.002
0.000
4- o.O(j3
( )
+
0.037
— 0.039
— O.<i0I
I
"•351
- 0.21U
— 0.0 J 2
2
-4-
0.127
4- 0.012
— o.O(j6
3
4-
O.OOI
f- 0.003
I
-
0.070
"" "Y "
+
..846
— o.2f.6
— 0.024
I
—
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