Dissertatio de motu apparenti stellarum fixarum ex aberratione et parallaxi annua conjunctim oriundo. Cujus partem quintam consensu ampl. fac. philos. Upsal. p. p. mag. Israël Bergman ... respondente Georgio Linck stip. Hammarskjöld. Ostrogotho. In audit.

H

D1SSEETATIO DR

MOTU ÄPPARENTI STELLARUM FIXARUM

EX ABERRATiONE ET PARALLAXI ANNUA

CON_{JUNCT IM OR1UNDO.}

CUJUS PAR TEM QUINTAM

CONSENSU ÄMPL. FAC. PHILOS. UPSAL.

p. p. Mag. I S B AEL B E R G M A N ASTRONOMI« DOCENS RESPONDENTE GEORG IO LINCK TST I P. HAMMARSKJÖLD. OSTUOGOTHO^'

SH AU DIT. GVST. DTE XXXI MARTH MDCCCXIX,

IT. P. M. S.

U P S A L I JE

KONUNGENS

TROMAKT* KAMMARRÄTTS RÅDET OCH RIDDARE!*

h6gÅdLE

herr

GUSTAF

NORDSTRÖM

SAMT

VALB O RM

fru

BEATA

NORDSTRÖM

född

KLÖFVERSRÖLD

Vordnadsfullt tillegnadt

Rf

) 33 C

i zzz_{fin Lat. cos P} {hfin E -f-rcot E} _-

fin P(bcos

E

_{- v}

fin

E)

-j- be cosdsecd'fin {P 4• dfi; ponaiur, ut

iupra?

b tmg u := ~, erit

v

i rr_{fin Lat. em P. b cosec u co?{E-} ti")_{- bcosec}

ufin(ii~E)finP

-4- bscos d sec d' fin QP _4- d')

rs b cosec u _{ffin P.fin QE-} «) _4-fin Lcii _cos P _ccs

(K

_-

nfi

-f- be eosd sec d'fin (P _-4- d'); denique,

fi

pona-fin P tang P

cur _{-tång a e= -— ■— = —t—,utinJ.Y\,}

fin Lat. cos P fin Lat.

b _{fin Pcos(E-u}-a) be cosdfin(P4-d')

^

!->

_{:(§. XIV).}

finufin a ccsd

Similifer fubftituris valoribus _{qvantitaturn cos C, cos P}

& AC, erit cos _{(C - P) =s cos Ccos P -4-fin C fin P}

cos P _{(J) cos E -rfin E) 4- fin Lot. fin P (bfin}

E-f-r cosE) S/[ffi cos E - r_{fin E)2-{- fin'2 Lat. [b fin E 4-rcos}

_E)*~fi

c& = cosP _{(b cos E-vfin E)4-fin Lat.fin P(bfin E]-{-r} cosE)

- be

cosd sud' cos _{{P 4- d') ss}

- b cosec u [cot P fin (E -u)

-fin Lat. fin Pcos ^E - u,] - be cosdsec cT eosQP d') Szt

cos P cot P

fi ponatur tanga' =—, ut jn §,vi.,

fin Latfin P fin Lat.

a~-bcosec_u\j-cosP

coseca"cos{E-u-{-a''y\-hecosdsecefcos(P}-<£"), feu

u=sbcosec u cosPcoseca"_{cos(E-u4-&")-becosdsecd!cos[P-^-df)9}

unde

) 34 (

#

&cos K-ii-¥■a") _cos.P\ be-._coru-coi-i P -+• d.)

(IV) a = _cos"fr~rr—T~"_Deel 7 H~7

fnu.Jina ■ ~cosd cos Deel.

Valöres angul i _E, ubi l'\ T_r

f

8c di' funt maxima

ve!

rr o in hac & XIV. _§, § inveniunrur ponendb

differen-riales harum _quantifatum & tpfas _{qua r}rita_{fes sr:To, quo} fa¬

cto, därur _{angulus E, ubi a',}

_ü

_8c cx funt rrro, & ubi

a' funt maxima _{per aequationes quarri gradus; ubi} ve-ro ^ & ex! funr maxima, _p:r sequariönes fexri gradus,

quas omnes aequationes, cum" longiores & rrsöatu

diffi-eiliores fint, quam ut111o-res, ad ferre fuperiedimus.. XVil.

Si _{parallaxis haberetur confians, esfer p pro rr} fub-ftiruendum in allatis vaiorsbos _{quanritatum ex 8c ex!.}

Aiia v.ro 8c _{exp-edirior eft via in Hoc} cafu

d

& ex!

in-veniendi.. Sit _{nempe, ut äurea, ABDE{üg.6 figura} va-riarionis, quae nunc eriam eric circulus, fpe£latori ycro terreno _{apparens ur e11 ip(is f§.} XV.)-, _q,uae fit BPEr £ntque BE, AD, HT, FG portiones paralleli eciipricae

circuii _{latitudims, paralleli aequinoctialis 8c circuli.}

decli-nationis, _{punctum C transeuntiumAT locus ftellae}

appa-rens _{pro cerco tempore- tunc en t} angulus NCE _r=E-h_å\ tång P

angulus HCE. rr: P, MCE= are (tång rr

=-jlllf J-j(XL* (g. W.J 8c NCM = E + n - a , unde s gfin (É -f- a - a') ßn P (v) d rr;. : ;—— -q- be cosdseed'fn{P.-f-d').. fin a ' . cot P

Si _{porro' angulus BOS [rr arc\tang rr 7—7—.)] (§.}

hit Lat,

) 35

C

VI.) ponarur, srantea

,~a',

erit

NV—fm

(iBo°-E-

a-o")

gfin(E 4- a 4~ a ) cos

P

fm (E 4- « 4-

«'

)

&

c» ~ ~ //J

17 Jul _a

— be cos d secd' cos cP +

<0,

uride

gfin\E-\-a-\~

d')cosP

be

cos

d

sec

d

cos

(P

-4-

/-)

^ ^

ym o" fox Z?m/. rox

Deel.

Hinc _{patet, esre}

_S

_maximum,

ubi

efl

fin

{E

-f-

_{a -}

a)

maximus, feu E =z 90° + « - .c,

J

s o,

ubi

eil

be _ßun cosd secd' fin (P 4~

d')

E ~ a' — a - nrc [fin— — —

],

*,/*

P

J

a' maximum, ubi efl: E =

90°

- a-

a"

&

=o, cum

efl:

be_fina"cosdsec

_d'

_cos

_{P

_4-

*0

E — are [/in_J ~ —

j

- a -

a\

g CoS P

J

pofiris 0, dt P

conflantibus,

6c

angulis E

4-

a

-

«'

&

JE 4- a 4- a"

pofitivis'"&

<^iSo°.

§. XVIII.

Sequitur, ut

exempla

quaedam

adferamus

variationis

fixarum, refpe£lo motu telluris

elliptico.

Easdem

fumfi-mus ftellas & easdem élongationes

folis ab iis,

ac

in

§.

IX, ut eo melius apparear,

quid

valeat

excentricitas

orbi-tae telluris in determinanda variatione.

Monendum etiam,

in hifce exemplis priorem terminum

cujusvis

formularum

(l), (II), (III),

<IV),poni

=

B

&

pofteriorem

z=z

C.

) 3« C

r* Fro Varia tione

Longitiidmis»

Ex, 7 draconis

Vtvxh.Longit.=3^9°48'44'» Longit.fiella? = 8*25*25'49%

Lntitiido z=: _{74°}7', .E =2 89°59's8"> *} = 0,01681 , 1 + « = 1,01631, _{i - c = o, 93319,} = 0V5, d = 5*i5°37y» P (i - é!2) E + d=z _{6^5°37r, r =} = i -«roj- _{(E -f-aj} p(i + #)([- *) r • —

—-, unde, fi _ponatur \^{Recos75°37'sO ₌

i _{4- * ro/75 37 3'}

p (1 -+- e) (1 - e) cos*A

tang A, erit r = _{~Rl ,}

&

le = _2.22557 lp _{= 1.69897} 4- /mr _75°37/3/' = 9.39515 4 /(i-f-<Q = o . 00724 4- _{IRzzziUang A=17 . 64072 -f-/ (i-ff)~ i} . 99264 Hang° A = g . 81036 4 2 lcos A ■=. \$. $9818 y, O ' ~l+ . -Ä = _{3 41 50} _ -2IRzzlrzzz _1.69703, i?/; Ib = i . 30643 & ex formula = — _{4. c. a.W = 10 . 30297} ltangu = 11 .60040 * = _{88 35 3 1} ? deinde ex formula

Rbfin (n - E) be cos d Rb fin \°2^2y'

(i) t

-+

cosLat, _{fintt cosLat. cosLat,fin} u

rf c* O ? f t

befinn 37 5 cos Lat►

r-co (ro *<«* VD O co f>- t*-Ö\ »h -^ •a-, \D «a» M OO QO C» OS v> 1-1 1 Cl CA O II II II II a j> + ^ 5 --» «o

+:

fö -5h vo O co eo cs t ' >/"N ~* O o

Csco

C m ir. O VÄ OS r-os er VO O Cl O II II

ÜÜ

II X CO -CO O 6 II II II II h* tj S Cl ^ s rt-t, ^ r< o . ,*. Q ~ 4* + 4* 1 CO -t-cl MD d es • -w O |{ I II II II «1 fi vo•c* VO s. * ~0 I O + e> il V) • >H G * r—I r-d S <D G O • r-t -H irH $-4Ctf > O Ph c* *q>cd sqj "3 us 3 • *—> <U "A r-II r-<T> lo ' -4 vs *"53 Ü-* *£ C/) £ o* cd o T3 c; 3 + r~\ na 1 rt II i V_/ II N ■-> O r- kr-UA r-+ + + « 04 il <L> SP r-V ro o to r-& > d K N **" CS £_ CO C\ O vo O ce vo Cs c» (i OS ov OS os I " O I H ^

(f>wN

Ofl t 1 — so oe CS O O SO ro co I M M Q * nr-' nv »-« 0C vo ••' rj" CO SO sä • as Cl v. VO Cffc CO o oöoe oo II II II II II II II IJ II I] " " 11 + + i ■-» 1 o + + II SS cl l <« IA "*

t^sO

cl so Cl OS fl os £ IX IA OS « O -• cl Z, 'S

<v~

II II II II il r* V lA r» + §P ^ ■2 5 Cl ^ ft* +: CS fc*0 vT c: •»«» »v» + s ti fel so ii Q

r*K

<") v =

) 38 (

bßnLat cos-[K- u) beßndfm Lat.

R_{finn R 2}

b_{fin Lat. cos}

_{88°36'3 i"}

_{befin i}

_cf

₂

_zfi

_%"

_{fin Lat.}

R_{fin ii R2}

Ib — i , 30643 Ib = 1 _. 30643

-I- tfin Lat.:sss 9 . 98484 *

-h'le

2.22557

-4-/ros&$,*36'29" = 8 • 3b54ö +lfi.i 1*12

35"

= 9 • 39512

-4- f. a. Ifihu _=0.00013 -4- lfinLat. _{= 9} . 98484

- ₂ IR ₌ IB _z=z 1.67686

- 2 IR=IC = 2.91196

i? = o",47_5i7 C₌ o",o8i65

A' =z B - C = o",_39352.

3. Pro Variatione Declinationis.

Ex. t5 y urfae minoris.

Quoniam eft Perih. Long. z=z

Longit.

fiel-1« =

4*ii°58'58".

d—

39°10,4"'

erlt'

Pofito

£=93°34-P O - «*) _

£ + <* = 3'44°44 '4 ■» r —

,_,«,/(£■+ rf)

~

jx (1 + O

₇

_{_~9;}

, fi ponatur

V(®e fin

4J°44

'4)

i -4- _efin 42°44

i4,/

i» (< + o (t -

6

t<u4^

r„ „ r.

= r= •

El

P°"°

^

flellae = 7V13V, £ = 93°34

52"

&

4^ \ t »o ^3 Ö il L k?ba a 4 4 »V *■>

4?*

I

•

CS

*3 II i"+

i"

1

5-i 1!

II

ii

im

OnM

i

O VO

VO

»3

uo

OV^I O

OCI

4-

OV

W

9V

M CO

oo

to

O

OO

W

Nj 4*

to

so

<-n

4

COW

^

W»

S<*

(?\

*4

Ö Ö-CO_oc_ o u> •

Oi

O V/l M O•" ^3 ^

II

+

s

^

vi

II

II

lf

o o so 4-Oa sa»II oo O CA 4-4 a 4-f «S--^ 4

4^

Ö Crq iÖ k °^

°^

^4»

x

ii

ii

ii

ii

"

ii

oo V© sc so UJ OO • •• • 0 =c VC SC o so OO 4k OO os — QO 4k"" >Ol 4k Tt o f1 4k Q uo 'S-VA ?2

c

Cr*vi 5 Cr3 Sl, rS t-» V nr"—», 4** a c§ ^*. V£o w o kO OO VC so cc 0 o K> v^l CO-vj vjk t>i o I-* 4 4 T /^A r-r>e 1 <-»k c* 4» ♦v o 4k 4\ n SP ocOO CA 4-II

II

II

sc- SDSCVCSC Ol tOoo Ov 4 ^-v» + II o 45* II OCAOso -J-CO KJ kO4-

A|

.00SC

l&Mfc

Måv

) 40 ( E = 159°24_{37 » E -f d =} MS*37V ₌ 3*J5*37V. p (i - *2) K1 -*-0(i - 0 1 -CCOS(E + d) _i■+■*/»

S5°37V

Vföfin 55*37V) P (i + «) (i - «) cos2A A i \ # ' V. / // Q , //• =ztangA,rzz — , « = 19 J>3? p = ₇₂ 34'12" (§. ix. 4)

J

le zzl z.₂₂₅₅₇ //? _{=5 i} .69897 lfin 55°37V' == 9•9»661 -4-/(14.<?) => 0.00724 -hlR= 2_{l lang}A22= 18 . »4218 -f-/(1 -0 — * •99264 l_{tangA -=s: 9.07109 --hllcos A =219.99402} ^ = 6 _{43 3} - ₂ IR_—Ir 22= _1.69287 fin Lat. tang if4733"

fang11 2= —_f tang a 2= _Ä

Ib z= 1 , 30643 /fin Lat- 2= 9_.961OO

-+• £, fl./r ■= 10. 30713 -hltang \3*47'3.3"=-9 . 3900z

Itangu ss 11 .61 356 - /J? rs

Langd!

2=: 9 . 35102

« == 8 _i9"

JE - « ₊ fl" = 89°56'2i" P -h d' 5/45 21"

Rbcos_{QE-u -+-} a")cosP _{be cosd cos (P -f- rf'}} £IV) os —

^ jj£C^_ßnu ßna" _cosd' _cosDeel.

(b ess I • 30643 ₌ I _.. 30643

«4-_!cOs(E-U 7.02602 _-4" sas 2 .22557

-4- lcosP = 9 .48046 H-/co/rf = 9. 98729

5 4« ( -f- c. a. _{Ißnu = o . 00013} -+• c. a. I_fma' _{zz 0.48441} -2 IR = IB zz i . 83487 B zz o"j68370 cc = B — C -4- r, o. Icosd' zz o. 0:067 -+- r. a. hos Deel.= i . 5374Z - ₂ IR _zz IC ₌ O . 76954 C zz _{5 j88z14} = ~ 5">'9§44

Ex _{figura 4. 5c}

_{conftruclione}

_{formularum (!')}

_{5c (IV)}

conftat, valöres poümvos ipforum / 5c oc tantum indicare,

hafee lineas _{fpeftitori terreno dextrorfum ab 5}

_h.

_{e. con¬}

tra ordinem _lignorum videri, 5c negatives, esfe

easdem

finiftrorfum ab 5 h7 e. fecundum ordinem fignorum

fum-endas. Formulae itaque allatae, fi quantirates

f

5c a

ex-hibent _{pofirivas, has}

_esfe

_{fubtrahendas,}

_{5c fi negativas,}

easdem esfe addendas longitudini 5c asfeenfioni re£las

ftellae, fignificant. Sic valör

ipfius

/',

in exemp'.o

1.

§.

XVIII, inventus, eft longitudini

addendus, unde fit

dif-ferentia _{quantitatum /'}

_{paragraphi IX.}

_(pofito

_{p zz}

_o",5)

5c _{hujus §. =}

_i",2789z.

_Qu

_od

_ad

_fignum

quantitatis

A'

attiner, ex figuris 1. 5c 2. patet , partes

figura?

variatio-nis, _{qua? fupra GK}

_5c

_de

_fitae

_funt,

_per

projeftionem

de-primi infra 5c quae

infra

funt elevari

fupra, ubi latitudo

ponitur borealis, qua?

5c

pofitiva

habeatur.

In

conftruen-dis autem formulis (V) 5c (II) plagas a K verfus A 5c ab e verfnsGpro pofitivis habuimus,ergo

fiunt

proje&ione

negativae; quare valöres

ipfius

A',

quos

formulae pofitivos

exhibent, funt fubtrahendi: quos negativos, addendi lati-tudini ftellae. Latitudo autern fi föret auftralis, manerent

portiones GL/f,

dDe

tirculi

aberrationis

pofitiva? 5c GHK>

dEe _negativa?;

_latitudo

_vero

_ipfa

_tunc

_{eslét negativa;}

i-taque valöres ipfins

A',

quos

exhibent

formula?

(2),

(II)

pofitivos,

femper

minuunr,

quos

negativos,

femper

augent

la-5 42 C

Jatitudinem ftellae. Variarioni o idern fignum eft tribuen-dum, ac id, quod formulae (3), (III)

proeftant.

§. XIX.

Ex antecedentibus conftar, figuram

variatiorils

esfe

circulum, etiem pofuo motu telluris

variabili,

modo

ut

parallaxis habsarur

conftans.

Redat

tandem,

ut

quaeratur

locus _{pun&i S' (fig. 2.),}

_refpe&is

_motu

variabili teliuns

ac excentricitate orbirce ejusdem & quod ad aberranonem & _{quod ad parallaxin.}

_{Fonantur irsque}

CQ

_{= x,}

_S'Q~y,

& ut antea CG—bi angulusque

BCG-S'GH—v,

erunt

p2{i - e2)2 ** + J„i _ S'C2 = b* 4- S'G> - b> 4-- —

(a)

%

(1 - e cosv)2 p (1 - e2) cosv y - bfmv = S'G. cosv — (b),

unde

J ' i - e cos v b_{finv {ecos v}

- 1) = p (1 - e2) cos v - y

(c

- ccos

v);

&, Ii ponantur cos v — u, p (1 - c2) — c,

b2e2u* - 2b7 ett* /£2 _ ^2 ^ ,2 __ _2cey

4-4- 2(£a« - ty - ey*)u 4- y2 - b* —o,

feu,

pofit/s

b2e2

—m,

2b2e — 11, &3(i - e2) + ra = d, mu* - nzz1 4~ (d - 2cey 4- e2y2)n2 4- (» - 2cy - uy2)u f- y2 -

b2

=0

(c).

Oritur porro ex aequatione (a)

g-fx* 4* yz " b2)n2 4* 2e (b2 - x2 - y2)u y2 4*

F*

-(b* + e*J = 0 (d);

) 43

(

& _quoniam,

_{exterminatione}

_ipfius

_u

fa£a

inter

duas

quascunque aequationes

quarti

Sc

{ecundi

gradus

An^ -}~ Bu* ~f" Cm2 -4~ Du E — o

Sc

am2 _{-f* bu -f-} c — o,

oritur haec inter coöfficientes A, B, C, D, E, a, b, c

sequatio

aic2C2 - 2ac*AC - la^cCE - abc^BC +

b-czACy

-fc4^2 + 2a*c*AE -

bc>AB +

ß4£2

+

^bcBE

- cib^cAE _- a^bcCD -f- ^abc^AD + b>cAD

S

rr O,

- a'bDE + cib*cBD + a*b*CE - ab'BE

4- b+AE + ac'B* +

a^cD* +

2a*c*BD

orietur, exterminato n inter

aequationes

(c)

Sc

(d),

asqua-tio duodecimi gradus inter

coordinatas

#

Sc

y;

itaque via

apparens tfellae

cujus

vis fixae,

aberratione

Sc paraliaxi

an-nua limul afFeftae, eft curva duodecimi ordinis.

Corrigenda.

p. pagina. 1.

linea.

f.

fuperne.

i.

inferne.

lege: p. §, 1. ji. u

S

= g cos

[a

A-

E)

—

l,

p.

16.

1.18.

19. f. fin (Lat. Ip he.

ßn

Lat)

_:

Rad.

feu

(t

Ißl

be

cos

Lat.)

fin Lat. : R, p. i\.

1.

5.

6.

f.

angulo

complernento

ejus,

cujus finus=fin Lat. taug.

differentiae

&c.,

1.

4. 5.

6. i.

css

pro ßn Sc ßn pro cos, p. 22.

1.

4.

g.

9. 10.

f.,

p. 23.

1.

1.

f. m _{pro n, n pro m}

&

_mutentur

figna quanriratum

zcosqi

zcot_{q. 2m2cotq uz , 2}

mrscotq.uz"

_■>

1.

₆

i.

projektår

_um,

J.

_2.