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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 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- 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 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) 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 ) + .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 (,■', -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 \ 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 ('/((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'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 ilependinK IIANSKNS 1.1 N.\K TllloKV. tlK'H put .; -[■ 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, ) — 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 + ) 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 90iii ( -j- .000 0197 >iii I -f .000 0013 >iii ( i7 + «> 3, ff + 'y) 4 <•/ -f A> a dice adoptcMl, Then -}-. 000 OOOI sill (6//-]-") F, — + .000 0002 COS (— 3 fi -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- — .001 3O .-in ( 3 V 4- oxj — .000 16 sin ( 4 '/ — «^* — .000 02 sin ( 5 ec<.]i(i uieth*>fl ot" coniinitin;^- it was as toHous. li"t lis put : — ti. 1.2 sin 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 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 .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 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 iii)^rniiciitl\ 'nvi'ii jrails to a ruiit" — 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 \\>, 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^.: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 coi;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- 2 TkANSFOR.MATIoN nF HANSEN'S I.I NAR TIIICORY. TaiiM". I. — I'llhli' (iJ'ik'^:, (('■ -( 'oiitiiiiic .1 = ("'':)■ f- (1.007 (/;.!:!' iiiA:.)' + (1. 1114 :'33 3 '■' - 3 '■' 2 - 1 3 -I 3 — 2 4 — 2 -3 I -3 2 -3 3 -3 4 -3 z -3 1 — 4 2 --I 3 -4 1 — 1 I -f- . 007 + O.OIO . I 1 - 11.(131 f- 0.052 -f . j I 2 1 + 0.007 • — 1 {) r 0.21)() — 0.013 -{- l.>.(-lo2 i» ■i- 0.31(1 — 0.01;.) + 0.017 1 ( t + 17 SWj - 0. 3S2 .i- 0. OIlJ 2 ( t + I). 25') - 0.044 + 0.005 — I — 1 _ 11. 564 — 0.O2I — . i.io; - 0.001 1) ~ 1 - 1 1 .400 - 2.7"; - 0.052 — 0.002 1 - 1 — 121.33? - 1.631 - 0.113 - 0.002 2 - I - I .')iii - 0. 163 - 0.042 3 - I - o.'\37 — 0.012 - 0. )OI — 1 — . OOl) +- 0.006 - 0.(JOl -2 - U.147 +■ 11-343 - o.(,'i6 I -;: — 0. 5')2 H 0..176 - 0.014 2 -2 — o.oSi t 0.074 - 0.004 3 — 2 - 0.006 + 0.004 ij -3 — 0.007 -t- 0.012 -*- 0.001 1 -3 -i- ii4[ + u . ( J 1 ■; -f- 0.002 3 -5 -f- + + O.OO.', + 0.001 •r o.;;o2 + 0.001 — 0.107 0.03-* — O.O! 1 - 1.1.003 0. 272 0.410 - 0.014 0.123 — 0. 2(J1 — 0.007 0.00() - . OOl) 0. (K)2 ^U l».Ot)*i +- (-'.001 I .OI)2 + 0.2iO - 0.044 3.151 -i- 2. 70S - 0.041) + ( ).oo3 0,621 + I.27.J - 0.017 -^ ( ).ooi o.otS -t- o.(,(, 4 ■ 1 4' I 1 . 5irt 0.118 o . 006 : u. 0.2i)6 o. 125 0.010 0.016 o.ooS 4- 0.004 4- o . 002 4- 0.054 -i- 0.023 J- 0.002 — 0.066 4- 0.672 t- 0.917 + 0.323 ^- 0.02S + 0.036 f o..)24 — l7-'il<' — 46.370 — 12.420 — 0.665 — 0.033 — 0.002 4- o . 00 ( f- 0.07.) — 4-311 — ? 544 ~ 1--^3J — .>. 1 12 ~ 0.006 f O.OO) — 0.254 — 0.406 — 0,15s — 0.010 — 0.012 — 0.022 — O.OOIJ 4- 0.035 4- 0.269 4- 0.260 ■^ 0.0S5 -(- 0.010 — 11.001 + 0,013 + 0.066 -t- 0.077 -r 0.035 -t- 0.006 0.020 0.246 0.259 0.094 0.013 0.002 o 026 o 031 0.013 o . 002 0.00 1 0.0(,)2 0.002 0.001 0.010 0.039 0.040 o.oiS 0,003 0.001 O . (.KJ4 0.007 o . 004 4- O.OOI TRANsR,...M,MU)X,,FllANSl.:NM,rX.\K lllKcuv. II /)Z I'OS. si,, COS. •J-''- - 2 '-' M 2 — I t- 0.002 — 0.0115 — o.ooS 3 4 — 1 4- o.oijS — O.ooS — 0.002 — 0.004 I — 2 -H 0.015 — 0. 14(1 — 0.004 - — 2 — 1 . 0(^2 + 1.71/) ■)- 0.024 3 — 2 — 0.002 + 0.S05 +• 0.026 4 — 2 + 0.010 - 017 + O.ooS ' -3 + . 002 — O.ooS t- 0.001 2 -3 — 0.044 + -(i + O.OOf) 3 -3 — 0.041 + 0.05S -t- O.ooS 4 -3 + -4 -t- 0.020 — o.oSd +■ 0.001 I --4 + 0.214 - •■^34 — <">.02S 2 -4 -I- 0.22.8 — 0.628 — 0.025 3 -4 - . o6f) -(- . uyo — 0.006 •1 — 4 — . 005 + 0.004 -5 • — o.oof) — o.uoi I — 5 + 0.021 — 0.1(j2 — 0,011 2 - ; -1- 0.030 — 0.07a — 0.007 3 -5 o.ooS + 0.010 — O.ooS — 0.002 I -f. 2 -6 — o.oof) . 5 '•' - 5 <■' -4 -4 -4 — 5 — 5 ~5 ~"5 -6 — 6 -6 -6 COS. sill. I'OS. ■ (,<.<■ — O.OOC) — . 006 • — 0.010 — 006 • - 0.002 - O.OOI + 0. 026 -■(- 0.003 t O.OOI 0.056 + 0.0()|) f- 0.045 f O.OOI O.IK); + 0.042 t- 0.042 0.007 — O.OOI - O.OI 1 . + 0.002 , , 0.006 + 0.012 f o.c)04 , ^ 0.001 + 0.003 + 0.005 0.0(J2 +■ O.OOI - - 5 — 0.002 — O.OOI 3 — 5 — 0.004 r- 0.01 1 f O.OI 1 0.003 4 5 6 -5 — 5 - 0.01 I O.ooS -r O.02S + O.OlS + 0,004 + 0.022 + 0.01, ■(- o.(:o2 - 0.005 0.0: .3 2 -6 ■T 0.009 — 0.013 + 0.015 .1 -6 + 0.2S5 - 0.652 — 0,50s 4 — 6 + (J, 53S — I.0S2 - o-ri'j 5 —6 -f- '5-33 4 — . 6 1 M — 0.35 1 6 -6 4- 0.034 — 0. 13S — 0.079 / -6 +- . 009 — 0.012 -■ 0.006 + 0.002 ] - — 0.001 3 -7 -t- 0.037 — O.oSS -- 0.070 + 003 ^ / + 0.0S5 — 0. 176 -- 0.126 -^- 0.005 6 / 4- 0.061 — 0.116 — 0.076 -i.- 0.003 / -(- 0.016 - 0.029 — 'l.OI- / / • — 0.002 — 0.001 3 — s — 0.007 — o.f/05 4 -S . — 0.016 ■ — 0.012 5 6 -s -8 — . ! 1 — 0.002 — 0.007 — O.OOI -4 -4 4,.. -61,/ 1 -6 2 —6 3 -(> 4 -6 : -6 3 -7 • — O.OOI — . 002 O.OII -t- 0.035 + 0.02S 0.016 ■t 0.03S + 0.027 0.005 -r . (JO<) + 0.006 • +- 0.003 - 0.003 • t- 0.005 4- . 003 — O.OOI — O.OOI 003 - 0.017 -- 0.030 005 - 0.019 — 0.025 001 - . 004 - . o( >4 • + 0.002 + . 00 1 - 0.002 - 0.004 - . o(j4 - 0.004 84 TKANSFOR.MATKiN OF IIANSI.N'S I.INAK I'l I i:t iF< V, 'Alll.r. II. — I'lhir't 1,1(1 jKirls iif IIaXSKn's /\r//ji/ir I.iiliilihlili\ ir'llli the Cnrljiciciih (if / III' ('iii'(ii((l((l Liii/i/ih((l(. ('•■A')i (lis X iS> («.';)• X {iii\z)'X ('■•A'h Suiii. '•J ^ 1 1'liiis in — . iiplir I, oil- .S- ■ ,nilii3 • ■■^T -t- 10. 157 • "34 -f- :,(...|2; .ooy + S.'i2; .iK-2 ^ .f.65 t + .041) + . "03 + .011; ■ * 7- -i ■ .410 .027 -r .103 . + .407 . -(- .I4f' . + .012 , + .(CI + .004 . -t- .003 -f- .C04 + ,(.03 .015 - .13.1 .i"il — .( iiS .fOd — ■ 1 3') a- .013 . 004 4- .0.^1 '"4 — I .f (15 rj(, ~ 1 .1 ''7 )'I4 - 1 . 5711 l'(;2 - i ■:-<") 0()() - . If s - .5 -I- .010 — • 292 J- .DOI - ■ "f'S - .00() _ .001 • 52f' - 2 • !<07 004 007 uof) (107 no 00 4- + + 4- 4- 4- 4- 4- 4- + 4- 4- 4- 4- + 4 22(137. 150 4- 7fiS..'!;S ,619 4- 36. 1 12 .0(15 4- 1.1)31 .005 4- .113 007 • 003 • <'3>) •551 7 . f)f)h I0().qi)8 6611.852 I 4 S , I )oo ')-7"J 670 .047 .003 .003 .065 1.1^4 7.5"7 .192 .014 .00: .01? .078 .04S .003 • 039 .522 I m'h) 3(1 . 1 •'4 12.3^. 36 .311) S.J04 •644 • 047 .(103 .003 .063 •3S4 .18S .434 • '57 • 014 .001 .004 .003 .004 .003 .015 • 136 .01 1 .136 .013 .oiS • 217 '•'44 • 031.1 K446 i.SSf) .226 .024 .002 .005 .071 ■975 4- 4- + 4- 4- + 4 4 4- 4- 4- 4- 4- 1- 4- 4- + 4- 4- .015 .230 2 •535 • iSS .013 .oiS •177 2.^21 28. 559 24.452 2 . 926 .292 .021 . 002 .005 .071 •949 e e e t e TRANSFORMATION OF MAXSKX'S I.rNAR T,„:oRV. 8; « t! ; X s s 2 M - — I () 2 3 4 5 r. — I () I 2 3 4 5 6 — I o r 3 4 — 3 -3 -3 -3 -3 -3 -3 -3 -3 -3 -4 -4 -4 -4 -4 -4 -4 -4 .rS„ — .013 + .137 + 3-3o ('■■A')i - I >n\: X Ri.i s Sj (>n^z)-X U-.K)' [n (! ;f X (■■. .di T<'riiis ill — c Kcliplic I.on- .Et'c giliido. .5 '% I — I + 004 4- 001 . .1 i 4- .005 4- .005 . o — I — (tf)7 + Olf) 4- . oof) 4- .002 4- ooS . . 1 . . 4- .025 4- .071 1 — I — 144 4- (lOI) - .074 + .00; 4- «)S 4- . (X)3 - .11)') 4- .oSo 2 — 1 + I lO) 4- 016 -1 .324 4- . 005 - (MJ( 4-. 010 - .l(,S - .07S 3 — I + . 123 4- 008 — .444 — .001 — 002 1- . OOt) — .307 — .304 A — I + . OK) - .065 . . . 4- . 002 - .053 - ■053 5 — I — .007 . — .007 — .007 I — 2 — Of) 2 - .001 . - .(X)3 - .003 2 __2 + Olf) — .015 -1- . 002 . — .002 -f- .001 4- .005 3 2 + 003 • .(X)7 • ■ — .001 . 005 ~" .005 . 2 ( 4 007 , , _ .007 .007 4 -)- . "34 4- .01; — .001 — . 002 4- . 046 — .068 I 4 - 007 4- .017 . — .OUI -4- .009 4- . 009 2 4 4- .002 4- .002 4- .002 3 3 . + 002 . 4- .002 -t- .002 2 3 4- 020 . , — 00.| . . 4- .Olf) 4- .028 I 3 + ■ 014 — 190 - . oof) — .ooS - 010 —.002 - .202 1 . 402 O 3 -+- I. 032 + 025 4- .283 — .032 4- 001 -.oiS 4- I . 296 — 2.153 '' I 1 — 024 — 1S7 4- .44f) + . oofi 4- 010 —.035 4- .216 4- .064 n 3 — . ooS — 022 4- . 0f)f) 4-. 001 4- 003 —.011 4- . 029 -t- .029 3 3 _ 002 4- .ooS — .002 4- . 004 -f- .004 4 2 , + 002 . . 4- .002 4- .002 3 2 +- . 001 4- 023 - 003 4- .026 4- .031 o 2 + . ooS 4- 2.JI - .003 — .ooj - 02S , . 4- .2f)3 4- .425 I 2 -t- ng4 - 4 504 - .117 — .UOI - 040 . . - 4-577 4- &.3f,3 I- (.1 2 + 23- ()()0 4- 342 4-3 .794 4-. 016 4- 024 4-. 003 4- 27.839 — 55.262 I- I n - . (.76 — 4 450 + <) .62S 4- 0=2 4-. 005 4- 4.551 — • 175 2 2 — 275 — 5()4 4-1 .472 4- 003 + , 002 -r .()3S 4- .561 . 3 2 - 02 S - 044 4- .167 • • 4- .095 + .095 4 2 — 003 - 003 4- .OI? . . 1 4- .012 4- .012 5 2 4- .002 . . * . 4- .002 4- .002 2 — 003 . 4- 00 . . — .002 — .008 1 — oOvS + 113 4- . iiuf) 4- (JO ■ 4-. 002 4- . I20 __ .07? o - 5"5 — 013 - • 044 -^.034 4- 0jX («-l:fX ■ ■ 4 s A'' -' s,, *} Sum. i-:c ■fiins in ipiic l.oii- ^ ^ u- -0)' (<•. ,c)i-i Ri,, /S, ( '>:■^^ (Ri,:) {e,g\. Hilu.k: i '' •»■ ,, ,, ■\ : I ~~ .(X>2 • , ■■ • -f- .<302 • .000 + .007 2 I _ ,()('2 • — .031 — 2 () + .022 .0.12 — . o< l.( - .004 — 1 o 1 2 3 o <) o o + .006 4- 4- + 4- 4- •"S3 •"33 • '/'5 .oSo -*- .01)4 .010 .'X)2 .Oil 4- . + . 4-. . + .022 .O-Q •977 • "35 • 97f> 4- 4- 4- f -f- .022 • 3^'9 ~ 1 . 2'.I3 - >; ' I7.''"I T,-' J .235 r ec 4 o 4- .006 ■ +• .003 . 4- .083 4- • "■S3 ' -3 . 4- . oof) + . o(j6 — I — .wjS • _ .(XJI ' 1 — I .107 . 0(.»9 . oog — I 1 ■ ~~ .on — .llS „ .IlS — I 1.053 - .082 1 . 16; I "''(1 1 ° I — I — ."5S — O.OqC - ■045 • 4- , 00 1 — 6.7()S _ . ],' — I — ■"47 — • 717 + .<-■( 2 — . 'J07 4- .07 1 2 3 4 () . 704 •549 -\- . 00 1 — • ^")7 — 122.032 T — I — I + + .<)2(J .002 - — 'JCI 4- .055 .012 . -f 001 — 6.630 • 53; - S.24'j r ■ 572 (' — I .oil 4- .002 5 — I — .IW3 ■"39 "" •"39 n _2 - . 00 1 • 4- .002 -v .003 .oor 4- . 003 . 00 1 o " .010 . . 4. .oM2 ,. * " - .IXJI .(X)I +• . 00 1 .001 + .040 .0(JI 3"- ■3„/ () • — 2 — .001 ' .002 , — .003 — , . 003 I — n 4- .01(1 - - — .013 4- .003 _ • 035 ' 2 rj + .00; — .lint — . oof 1 .002 H" .270 ! 3 — 2 ~ (){)I + .i)i6 - . -j- .013 4- .'127 + . 1 5" .024 4 — o + .of.>S . . • ■ 4- .007 4- .015 + 5 — 2 4- .001 ■ . -1- .0(51 4- .002 ^- .002 i -' -3 • — •"05 . . - • 4- .002 _ . "03 — .003 o -3 • ~" ■■'72 . , • • ■»• ■ "1 7 — ■ "55 __ ■"57 ' -3 — 001 — • '71 . . — .oSo .0()2 . ••154 - 2 -3 -f- (>()() ■— .02f. . . — .fyrn ^ .02') -V .01 I 3-'13 - e 3 1 — 3 + 01 S ~ .176 . . ~.rf^, — ■"74 — .226 ■f ■395 ^ ! 4 -3 + <"'3 + .0:2 . . — .-,"0 1 — ■ "45 — .UI.) — .1 01 • 5 --3 4- .003 . - — .ooS — ,005 __ . 004 ; o -4 • i — .005 - - 4- !»! — . 00 ( — . 004 ' — ) — "13 • ■ - - 4- .005 — .ooS _ ■ "74 ; == -4 + ."01 . . . - 4- . fXJ2 4- . 003 _ . 226 i 3 -4 — .013 . . • - — .004 { • — .017 -f .061 i ♦ -4 + .004 . . . - — .003 + ,001 4- .004 ! 2 ~5 * ' * ' * — .012 3 — 5 .001 . . • • — .001 4- .(X)6 88 TRANsroiorA ricN oi" Hansen s ijnak theory. 'rAiii-i: II. — /'//'• Mniiii's Ldi/i/ihii/i — ( Vi]itiiiii<-«i. .T A s s. « 1' : X Ui<\: •X Kniz^-K Sum. T E.:li urms ill ptic u, :i- i C (',.<). — I K„, fS, (<■ .<) K,.s it,t\t fi ituilf. £ 1 rj -f- LI ,, ,f ,, ,, ,, 2 2 + .rx)r , . *, . + .U I . , - .002 4- .fA)2 . 1 -I , , + . 01 >6 . + .001 . . - .007 4- . 007 O — .02) + .042 4- .010 — .001 — .002 - .023 4- .073 , I + .Ot-l 4- . o( )3 — • 243 - .006 — .001 — .0( 3 - - - .lift 4- .541 . 2 + •"3? + .042 - .063 + .002 — . ooO . . -r .010 + .010 , 3 4- ."04 + .01 '3 - .010 — .00 1 - .003 - .0(55 , — I , , 4- ..""3 -r- .003 4- .003 . I • • + + .002 .(103 + .002 .035 — .IX I . • ■#- .000 .037 4- +- ■049 .061 ■ 2 - .003 + . 0(I2 4- .006 • • -»• .005 4- .005 : . 3''' -,..' ! 2 -r- .003 , — .(^02 • -- .0«>l +- .001 , 3 - .035 - .uol . . — .')3fi - .036 . 4 — 007 . ■ . — .007 — .007 I — I -^ f)03 4- Ot)2 — .(K)2 . . ^ .003 4- .oo-^ . 2 — ! — (125 ■i- OOI) — ."T02 — .002 , I 3 — 3 , -u 001 022 + .001 .022 — . '.<:■% ■K .003 4- 4- .002 •039 2 -3 . — "3- • 4- .02S — "'-2 — .006 — .356 , 3 -3 -4- 007 — 032 4- .i - .032 - .(140 i' I' 4 -3 -r 005 — 037 + ■ o'j; — .027 -.005 — ■'^37 - ■293 5 -3 -r no I — 016 — .013 — . 'JOI - .020 — .052 , 6 -3 - (-03 • . - .002 - .005 - .(305 2 -4 4- 001 . . — . 002 — .001 — .001 . — I -4 • + 4- OtI ^ 7- .017 •174 — .006 .nil — . 006 .02S • I -4 - 'X)I + I 7S5 4- 002 — 1 •495 -«- .2f>I a- 1. 177 .-■' 2 -4 — 0,3 ^ 2 050 + 025 — I ■344 Hi- ■725 4- 3".7fi8 ,'- 1 3 -4 — :r,r •)- 2 242 — 242 4- .i/)5 — IX) — 2.7.J3 -r- 3S.426 (■ 4 — 4 — if>3 4- 2. 117 - 164 4- I ■434 - OfJ. - 3.217 -i_ 1 3 . 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() 3 i-. .,,2,) — . ,„ 12 (-- .027 — .(■(12 4- .,125 4- .oil) 4- 0.02 ' 3 — .14^ ■r .1121) — .116 + .02a — .(i5S — .,i()I — U . ()i) 2 3 - .021 -• .001 - , ,122 + .om — ,012 — .012 — ( 1 . 1 3 3 — .,)02 - .'< -L- .2)1 -t- 0.2(J 1 2 — 3-5115 h .7l,S - 2.797 -U .6(13 — 2.1,j4 — 2.1,)0 — 2.1,) - 2 — .(,,11 f- • 'M3 - .55S + .232 - .32(1 - •313 - 0.31 3 - — • "''3 ■- .063 + .(141) — .023 — .023 — 0.02 4 2 — . ( 1, 1') . - . (-K.,6 4- ,(105 — . ( 'O I — .0( ,2 , , 1 1 I — . ,-ji I'j - .IJlllJ 4- .(1,11 — .('U^ — .1105 — O.OI 1 1 -U .085 - .ikiS 4- .077 — .(112 )- .06 = 4- ■054 4- 0.05 - I + .,)(>(i ^» * . 4- ,<.i()0 - .(-1(12 -t- .004 4- .005 . 3 I + .OlJl -h .UOI . 4- . 00 1 , lOO 1 RANSl-ORMATION OF T W'SKN'S M'NAK TIIFOkY. Taum-; I\'. — Thv M«nii's Lntilndc — " -iitiiuK'd. -4 -4 -4 -4 — 4 sin I sin (f+i. .()()2 .oio ■"33 .(H)4 .OOl , ( >( t I .OM3 sin A (X)4 "54 u()3 A — sin A A llaiiSi-tt. 0()2 — ord 4- (K)2 - 02i) -t- (1()^ - o;o f DiS + OOl + 001 (X)I - 003 -T- (K)I - 003 + 002 - .002 .(loS .024 .068 ) . ( 10 1 .002 • OOt /h-lllHlhiy PehiHiitiy (2). 002 002 ofi \ -t 006 (- O.of) 3 4 ! — 1 2 -I 3 — I 5 — 1 001 nil ous THIS r 2&S - SiS + 2t^2 021 (_»|2 (H)2 003 + 003 137 002 + + + 001 . + .001 .oil + .()(> \ - .007 — .1x15 + .ooS -f .013 + -(- . oud + .006 + . -(- .001 + .001 + .ooS — .001 + .1-11)7 + 131 - .023 -r . loS -t- 816 + 1 .1110 — 2 . Sofi — 252 - 6. 05 I - ( ■ 3"3 — 02I - 1 . 000 — 1 .021 - 002 - .117 - .11 ) — - .012 — .1)12 — — .o.)l - . ( )0 1 002 + , 1 )02 (K)f) - .004 + .002 ■r ,003 - .003 - .007 - .n()7 — — .(M-)l — . 00 1 — . 006 .013 .007 . Ol) I .1K)I • " 33 2.fiJ7 6 . 297 I .oi.S .119 .OI2 .ooS 0.01 0.0 1 0.01 0.13 2.70 f).3o 1 .02 o. 12 O.OI .006 — I — .oi_)5 — .005 — .005 — .001 2 - . \\U - .llf) - .Il(. — .OIJO — 0.09 3 — • o>5 — .015 - 015 — .009 — O.OI 4 - .001 — . 00 1 — .001 — I - . 002 -t- 003 + .001 -)- .001 — 001 I — 1 4- . 1 11 - 052 -+- . of )0 4.- . 0()0 4- .U21 4- 0.02 2 -I — ■ :74 + 256 - l.3'S - 003 - '. .y.\ - I..'^ci2 — 1 .50 3 — I — S.430 -f- 150 - > 'So 4- 003 - \.-zTi - I.3S2 - 1. 33 4 — 1 — .259 ^ o<7 — .242 4- OO' — . 240 - .239 — 0.24 5 — 1 — .034 — .034 — ."34 — .023 — 0.02 f. — [ — .003 — .003 — .003 . — I — 2 4- .IX) 5 + 029 + .034 4- ■ "34 4- ."25 4- 0.03 -2 — .004 + 273 + .269 4- . 269 4- .240 4- 0.25 1 2 — 1.85^ 4- 23') — 1 .622 - 01 )i — 1 .623 — 1 • 73'J — 1.68 2 1 + !<)<>. 4 7<> - 296 H- 1 99 . 1 So + 303 4- "'^■4"3 4- 199.277 4- 199.42 3 — 2 + "7-753 — ()(/i + 117.657 - 39') 4- "17.258 + 117.1SS 4- 117. 19 I 2 -(- 15.207 — "15 + I5.l<)2 — "77 4- 15.115 4- 15. 105 4- 15,11 5 — 2 + 1.531 — 002 -f- 1 . 529 - 010 + I. 519 1- 1 . 502 4- 1.50 6 -2 4- .141 + .141 - 001 4- ,14" 4- .132 4- 0.13 7 — 2 -f- .012 4- ,012 + .012 4- .008 4- O.OI ■- --...„ , ■_ TRANSFORMATION OF HANSEN'S LUNAR TIIF.ORY. Taiuk ]\, — ■/•/„. M„f„f',, Ltitifuilc—ContuniMl lOI ,C' sill I .siii( /+ (,j) 1 2 3 4 5 fi 2 3 4 5 - 3 4 3 (., - 2 O I 2 3 4 — 2 — I O I 2 3 4 5 — I o 2 3 4 2 3 -3 -3 -3 -3 -3 -3 -3 -4 -4 -4 -4 4 -5 -5 — 5 4 w' — 2 -3 -3 -3 -3 -3 -4 -4 -4 -4 -4 -4 -4 -4 -5 -5 -5 -5 -5 -5 -f> -6 I 2 3 4 5 6 4 -fit -6 -0 -6 -6 -fi -6 -7 + + + + + + + + + + + + + + + .071 ,,.iS4 S . 1 80 I.ldf) . 12S .1113 .003 • 334 .4111 . oh I .007 .t 12 .017 .002 .001 .l)(U .003 .010 .005 ,t)l)l . 03 .012 • "43 .220 • f'"3 .23(1 .021) .(X)I . 002 .U05 .oiS .of/, .031 • 5 .005 .001 >1U , > ( - s in ,i Ill'tSCIl. Deliiiititty De/ 11111,1V (!)• (2). ' ^- .010 ^ .010 + .010 : 4- 1* .010 4- O.OI h .oir> — .061 . - .061 ' .„ .07S (J . oS - 1 ( £ - *.v>s -t- .014 + 8.f;i2 4- 8.96S 4- (J. 00 . 1 r^ * .022 -t- " .012 — .015 \ + 7 •007 1 + 7.<)4f' -1- 7-95 "^ ' •«44 — .(XJ4 '< + 1 . 140 4- 1.0S2 4- 1.08 • ~ - £3^ — .oot ■ + .127 4- .100 4- 0. 1 1 ) "^ .013 1 + .013 4- . ou6 + 0.01 ~ .003 — .003 1 . 003 ."I4 "^ .J20 ■ + .320 1 1 4- • 311 4- 0.31 -Ul 1 ~ ■3Tf> + •3i)'> 4- .3(^2 4- 0.36 • "*" .fjfti + .Of.l 4- •043 4- 0.04 • ~ .'jr>- +- . < 107 4- .002 — .012 -f .012 4- .005 4- 0.01 '~~ .017 + .017 4- .006 4- 0.01 ■ .f'OZ ■+• .002 . . - -CjI _ .001 .002 '- .Q»>I + .oot . U02 — .001 .001 _ .014 „. 0.0: •'4? — ■»:3 — • 153 — •i'» — 0.20 . OI(? — . l.jj — .103 — .114 0. II .UI2 — - - .013 • 003 — .014 — O.OI • • — .012 .OI2 — . 002 .045 — . 4- 3.f'5 • 44? 4- .46a — 002 + .4^) 4- • 4'.4 4- 0.46 ■ 047 T- .045 .'02 mi + .048 .002 .005 4- .044 4- 0.04 .01 1 -i- .ryiff + .02Q 4- .042 4- 0.04 • 45<-' ^- .•If, -i_ 0O£ + • 517 + • 576 4- . 60 • 379 -^ .41c + .410 4- •SOS 4- 0.40 .050 + .053 ^ +- •053 4- .046 4- 0.05 .021 + .02; 1 • + .025 4- .02S + 0.03 .01. J 021 OOI . + .021 .001 4- .022 4- 0.02 .CK>5 ~ oyj 1 + .003 4- .003 .oOS -t- 075 + •073 + . t)6 1 4- 0.06 .0H6 + 091 + .091 4- . 07 1 4 0.07 .03; -t- 036 4- .036 4- .(124 4- 0.(j2 .002 4- txja 4- . 002 + .0112 001 4- .UOI 4- .005 lo: TKANSroRMATloN tU' IIANSKN'S l.LNAK l\il.'Hi\. '\\\u].v. IN'. — The M(i(ii/'-< l.dlihcl' — ( "uiitiiiia-«l. ^ ^' sin I sill (_/•+•''') sin i3 /; - sin ,} //,,„if„. • I) /J /K/iiiiiiiiy (2). -3 () -f- __ o O + — 1 IJ + II 4- o -t- — 4 — 1 — -3 — 1 - -■:!■ - t — — I — 1 — — I - — 1 - -I - -2 o *r — 2 -3 2 - 4 — 1 5 — I I — 2 2 u — 3 I.) ( + 4- + ) — .00: .0x2 .016 •T'/3 .013 .001 . I !'■ .423 4.<::-- ■ -'i -t'3l .tlOI .019 .47 — .o;3 - .0.34 ..-13 — ..-x)4 .ni2 .033 .002 .034 .2 5 -3 — .007 f) -3 - .OUI 2 -4 - .003 3 — 4 — .014 2w + (./ 1 I +- .oor • 2 I -t- .035 3 I + .004 I o -f- .002 2 o + .001 ■tw - 4 o 3 --I + .003 4 — I 1 5 -1 1 • • i .0U3 — .001 + . 005 — .030 — •055 — .027 — .005 — • 003 + .OK) + 2.415 + 3-017 -t- 1 . 204 + .210 + .02() + . "03 + .218 + •347 + . 102 -f- .031 + • 003 -h .0:2 + .024 H- .013 + .002 + .OUI + 4- + -1- .002 .021 .or4 .01)2 .054 .20S .030 .(,07 .001 .003 .UI4 .001 •('35 . 004 .002 .001 .003 + 004 013 oir 002 001 001 -f- -t- + -I- + 4- ■t- + + + + + + + + + .005 + .001 -t- -f .001 + .001 _ .001 + .005 + .001 + + .002 • OOt .005 .030 • "55 .027 .005 .003 .01.) 2 . 1 1 ,j 3.004 '■l'J3 •214 .021) • 003 .21.S • 34^ .161 .031 ■ 003 .012 .1124 .1.13 .002 .ooi . 002 .021 .014 . 002 .054 .2o3 .021) .007 .001 .003 .014 .001 + .030 + .003 + .002 .002 . o I S .01 1 .002 • 043 .if.5 . 005 .005 . 00 1 .002 .005 • 03- . 004 001 002 _u .002 005 + .005 OOI 1 • ■ + 4- -1- + .058 .05S .020 ,001 .00(1 2.S16 I.07.J .Ifi2 .012 .168 + .256 + .102 -1- .010 + .005 -f- . I >oS + .003 O.of) O . of J 0.02 01 2 • 32 n S.J t orj 16 01 17 26 10 01 01 0. 01 o . 02 O.lll o. )4 0.17 o . O I O.OI 0.03 I04 THANSIOUMATIUN OF IIANSKN'S LUNAR THEoRy. 'rAlll.K W . — 77/r Monti's Lllt'lllltlf — ('(intilllKHl. K g - 3''' o sin I sin ( /-fw) ^ f.i — 2 (.1 2 2 I 4~ 3 — 2 i — 7 3 4 5 6 4 w ■ I -3 -3 -3 -3 ■ 50' 3 -5 ()U — \ (,'' 4 -5 5 -5 7 (.1 — 6 (,)' ^" -6 -6 -6 -6 + + + + + 5 6 7 3(j + -7 j + -7 I + -4 5" 4 o 5 o o .003 .015 004 oSi) 0f)0 008 ex)4 004 .001 .001 .001 .005 .002 .031 .060 •043 .014 .002 .004 .010 .008 .001 .001 .001 sin /} + •t- .031 + .060 + .043 + .014 + .002 + .004 + .010 4- .ooS + .001 .001 .001 /^ - sin /i) .002 .015 • ! + 004 O81) I + Of)0 I — ooS ! - 004 \ + fx)4 — .001 .001 .001 .005 .002 .not .no I .024 .isr, • 137 .030 .r)04 .(«)3 .OOl) .002 — .004 — .006 — .003 .001 + .002 + .001 + .006 + ,002 //■linen. ft /^f/illlllilV (I). ' ft /'>2 — .004 — (->02 , , • •! -3 — 2 — .()II2 — ,002 , . . 1 — 2 -2 - .018 -t- .1104 - .014 ~ . ( >oS — 0.01 , , — I ~2 — .213 + .092 — . 121 — . 101 __ 0. 10 4- 2 — 0. 12 -2 — 2.I2S + t.S2f) — . 302 — • 277 — 0.2S -1 2 — 0.31 I -•J - .(Ji)2 + 3?-3'>i +- 34.301) + 34 . IM. + 34.2'| -(- 12 4- 4- 34 • 3') 2 — 2 (- I.()(JI> + 2(1. 23^ -t- 2S.225 4- 2S '70 4-' 2S.2(J 4- n --i- 4- 2S.23 3 — 2 + I . I(|i) ,.j_ ■ .S.)4 -+- 3^'iS4 4- 3 .o('i4 4- 3.07 ■)- I 4- 4- 3-"') 4 — 2 -t- •154 -r .129 f- • 2S3 -(- 271 + 0.27 4- 4- 0.2S 5 -2 + .01; -h .«)S 4- .023 4- oiS + 0.02 • • 1 6 -2 + .001 + .(X>I , , , . ' — 2 — 3 - .001 — .001 , , , , — I -3 - .ooS -f- . 1)04 - . 004 — 003 * i — 3 — .094 + .075 — .01l| — 013 — O.OI 4- 1 I -3 — . oOc) + 1 . 5 1 6 + I--I47 + I 452 4- 1.47 + 2 - 4- 1-45 2 -3 -1- .Of)t + I.?29 -\- 1.920 4- I S7f, 4- 1.91 + 3 4- 1.92 3 -3 + .0S2 + •'47 -f .229 4- '97 4- 0.22 + 2 -r 0.22 4 — 3 + .012 + .010 + .022 ^- Ot2 + O.OI 4- . 5 -3 + . 00 1 + . 00 1 i Aifamt' siiiu Paiallax. io6 TK w'^i-mni \ rid.' dt ii \nsi;n's i.i'nar iiiroKV. 'rAMM', \ / /" M'ini/s I'lll'l/ldl — ( 'cilltiinicd IXt • con /) a.w - 3M i> -» 1 -4 2 ■4 i - » -1 -4 I -5 2 -i !'■' - .4'.! •J - 3 3 -3 4 - 3 — 1 3 -■4 5 -4 J — 5 3 — 5 4 — 5 — (> .fU -!•') 3 + 3 3 - M)| on ^ "M (111 I I'.ll. IlilHSili s Sillf I'.n.ill.ix. IK>3 — <.<3 + DillJ •»- IK.S t- mil + oo| ■t- . . — .004 — - .■Hi|, - . ( ><> 1 - .I.I1J -t* .1)111 -t- .IJOf t- • 377 1- .1122 1- .577 -1- (i.dJ s- .231 ^■ .1)12 + • "31 + (ii)'J h .(102 ■+ .nol + •"33 H- (1(>2 •h ■ Of) 7 + ""4 . 03 1 ■+ III '2 '■ .1.1,5 ■(- . f- .i'u2 -t. - .1104 + . 1 II c + . . t . nil 1 + .oi).4 . .. -t- .1)111 t' .nil) 1- .01,7 (• .i'ii7 .1102 t . 1 11 )2 . . -'r .IXJl + .'.ml . . 4- .(KJl + . 00 r /) ■'illlll.iv'j /KAiiiiiiiy SIIU' »int' 1 ;ii.ill:ix. l'.ii:ill;iN (11 (-'1 OI)l "4') + 0()3 T 01 a + oil I Dill (1(1) 004 _ 11(11 J — tl(l| - IMiS + 373 •H S<)<) + 2fll •+- "43 + 004 032 4- Of(^ + "35 •r 01)7 (,02 1)114 1J02 1), I II I) xinc l'.ii,ill.i\. "45 +- i).i)4 1) t 1 )• (i.o; 07(1 ■f 0. In ■ (-- , 3 - 1 H- O.OI) (K)5 4- o.UI u . 007 - (joS — ( (12 (104 310 t- 4'W t lijfi •t- 01 (J + i)iti + 030 h (111 4- (J')4 - .i'"3 — .(11)2 o(;7 - .007 - .007 IJOI — .002 — .""3 0.01 (l.OI 0.31 (). to o. 20 (1.1), 0.0 "•" 0.0 4- (1 f- 1) h 10 I- '< t (1 I Oil- a 1 . _ .003 — .003 — ()()2 . 1 4- . ()( 1 1 -(- .001 f 002 . -t- (1(»2 - .0(i2 • . . + (,3s - .038 , . . - • ;<■"} - ■ 7"') - :oS5 - 0.71 <1 - 039 T- .-127 - .U12 — (HllJ — 0.01 II — 002 -;- (;ii2 (1 1 + , 002 •)- .002 ■+- 1102 . . I . + .UOI + ,tK)l + (X'2 . * • 37 (ill 2(1 i>(, 0.7 I U W'^l |>U\I.\ I |((\ (,| IIWSKNS I.INAU llll.oKV. IaIII.K \. — 'I'll, Muiiii'a J'didlliU ColllillllL'd. 107 I>(l_+ teas/) IVrl. — 1 I 1 + 4,., - a u' — 2 — 2 - — 2 - 2 t.* -4.,, t 1 -4 -» u — w' -1 I) - — 1) -t (J -(- — 2 — 1 -(■ - -I — 1 H — 1 1- ■ 1 — — I - - 1 -2 2 — 2 ,3'.- - 3 '"' ' 2 — 2 3 I -3 2 -3 — 3 -3 - 2 --1 ! ' -4 <.i ■( r a — Ut 1 I -3 1 1 I Inn ten' < siiii- I'.iialliii. . — .<«>4 ._ ,)3S .(Ad .. (Hi; . 1 ] J _. UJI) — • :^ — mil . . _ CHII t- . < inl t- . ( )i } 1 ■♦- 001 • • , •H , -^ .oil _ ). II Q^ < )