Dissertatio de motu apparenti stellarum fixarum ex aberratione et parallaxi annua conjunctim oriundo. Cujus partem quartam consensu ampl. fac. philos. Upsal. p. p. mag. Israël Bergman ... respondente Andrea Laurentio Linck stip. Sleincour. Ostrogotho. In

DISSERTATIO de

MOTU APPARENTI STELLARUM FIXARUM

EX ABERRATIONE ET PARALLAXI ANNUA

CONJUNCTIM

OR1UNDO.

cujus partem quartam

CONSENSU AMPL. FAC, PHILOS. UPSAL.

p. p.

MAG. ISRAEL

BERGMAN

astronomie docens

respondente

ANDREA LAURENTIO LINCK

stip. seeincour« ostrogotho.

in audit. gust. die xxxi marth mdcccxix.

h. a. m. s.

UPSALIiE

J *5 C

Jam fi t& pro

fun&ione ipfius

cos 2v

habetur,

valör

ipH-iis cos 2v & inde ipfius _vt ubi' eft _z vel maxima vel

minima, datur ponendo

dz

= o. Eft vero d cos 2 v

i-k* fin? (u-m)s 2 j i - k2/in2Cv-**)

dz=i £(

)

•

d

J

^

&

£24- e2fin2v b2 -f- e2fin2v i- k2fin2 (v _-«2} 1 _-?k2[1 _- cos (2 v_-

2?«)]

b2 -4~ _{c2fin2v b2 -f- 4}

_b2

C1

_{~cos 2} q 4- *. k2 cos(21/-im) ir - _j e2 _{cos 2v}

pofitis i -

jk2

= q

& b2

-4-

J e*

h.

e. i

-§

e*z=:i',

feu

q +.±k2Qcosivcos im

«4- fin

lvfin

2«)

r-\t2 _{cos lv}

2^4- ck2X-b k2S

%/(l-X2)

2r -e2x

pofitis tos 2vzzx,fin 2tn = j

&

cos zm = c.

Itaque

r(ar—£

rft*dx-(jxr-e* x^k

2 """T i~k*fin2

(v-

tn)

\

VilmX

)

>

4

£*-H«2/»JlT

~

(gg+ '*'*

-I-

*'J

«•'fa

(2r -4 Hinc

5 26 c

dz

Hinc _{ß ponatur} —- = o, oritur oequatio tiX e^f^sx* - 2k*rsx 2ck:r-cd2k-x _{-j- — —- -4- 2e~q -b} ce'2k2x VC« - *0 -J- «2^2J- y\I - x2J22 O feu _{ÄVX1 ~ ~ gx - n/}

pofuis 2ck2r 4- = h, 2k2rs=z g 8ce2k2s 22 «, und© gn ± ßj/(ga ■+- ^ ~ «a)

.v rar 2t; 22: _{; .}

g2 4- A2

Quoniam eadem operatio & maximum & minimum

cujuscunque fun&ionis quantitatis cujusdam variabilis

ex-hibetj & pra?f«ns quadtio urriusque capax eß, ex duplici

valöre _{ipfms x certo concluditur, unum dare maximum}

Sc alrerum _{minimum, praderrira cum in curva fimplicis}

cutvatörae non nifi unum _{maximum &} unum minimum

fetriidiametri esfe _{poteß. Duplex itaque hic valor iplius}

x, in _{expresfione generali iplius z fubßitutus, duos etiam}

exhibet valöres _{ipßus z , quorum unus maximam Sc alter} minirnsm _{femidiametrum, h. e. unus majorem Sc alter} minorem femiaxem novse _{ellipfeos indicec, necesfe efl^} unde etiam excenrricitas fädle datur.

§. XIV.

Ut _{jam inveniatur mutatio variationrs,}

quam

effici-«nt alldtac _{insqualitares conjun£lim, ßt ADBE}

(figg. 2 Sc

3.) circulus aberrationis, cujus centrum C, radius =£ Sc

g, ut antea, =» rationi excentricitatis orbitae relluns ad

di-) ä7 (

flantiam CSzu_be; fit _{porro G} locus flellae _apparens

in

cir-ctilo _{aberrationis pro certo tempore, AB diameter, dueta} per S, quoe femper e£t axi minori Orbits telluris paralle-iå _{{§. XI.). Ex antecedentibus tunc pätet, ii aberratio} fola _{egerer, ftellam in piano, eclipticae}

_parallele,

_peri*

pheriam ADBE inträ anni fparium rnoru apparenti

de-fcriptnram esfe , <5c, fi ftella firnul parallsxi afficeretur,

punflum G non amplius före locurn Hellas apparentem,

fed focum _{ellipfeös rKTH, in cujus periphena Hella}

fer-ri videretur. Focus _{hujus ellipfeos, iri peripheria circuli}

aberrationis fitus, qui vocetur f, ad eandem partern a

eenrro _{vergit, ac} aphelion orbit« telluris _a föle, «Sc axis

ejus major eil axi majori orbitae telluris parallelus {§.

XIII b qua re erit ad AB perpendicularis (fcucl. XI. io.),

Hoc fitu _{perpendiculari obtinente, quocunque It -11}a fir,

neque mutarur, fi ha-c aberratione etiam

afficererur,

quas

tantum focum elbpieos transfer*. -Cum _terra eil in

aphe-lio n, Hella in ellipfi parallaxeos in pun£to H apparet <Sc

focus f in B; terraque ubi deferipferit anomaliam veram aFJA{zzzESLzxsv)) punctum quoque H par. ellipfeos ad 5' transfertur, ita ut (it HGS'zzzv, & focus ellipfeos / a B ad G, ut fit angulus LSG _sz igo* ■— v —- are (fm

cos v- e

___

, -;)(§• XII.), itaque BSG = . . ,

V(t-zecosv-b-^y

y

*

cosv- e

do6 - art C fin )& fm BSG _{= cos}are

V(* ~2tcosv -4-e*)

cosv - c finv

(fm = -j ,)=s — .

Quo-V(i Itcosv) \\\-2 ecosv+e2)

niam porro SG eft ipfi TZ proportionalis, & haec, tel-iure lita in _{aphelio,=*« (i - t), illa veroa«6 (i -<?),erit}

) 28 (

bTZ

TZiSG::a(\-e):b(i-e):: a:b,SG == _a -=^bV\i-2ecosv+e2)) (§. XII.) & GO zzz. GS. _{ßn GSBz=.bßnv. Habent itaque}

anguli GCB & v eundem finum; funt etiam fimul acuti3,

fimul re£ti Sc fimul obtufi. Eft enim _{angulus SGO = /,}

quoniam funt SG, GO lineis VIyIU _{parallel«, & SOG} z=aUtf, _{ergo CSG}=90* -f- /. H inc paret, quamdiu. an¬

gulus 1 eft pofitivus, tamdiu esfe CSG obrufum Sc inde

GCS acutum. Tellus vero ubi ad N_{procesferit, fit 7= o,} CSG=_{90° Sc focus / in D}

apparet, quo rnomento itaque

v Sc GCO adhuc funt _{fimul acuti. Tellure aurem ulterius}

progrediente, focus / quoque verfus A procedit, donec

fit y =9o°,

quo fatfto etiam erit GCO —90°, _propter

ae-qualitatem finuum. _{Augefcenre dein} vf augefeit etiam

GCB; _{itaque hi anguli}

_Tunt

_{fimul acuti} , re£ti Sc obtufi„

Sc _{angulus GCB femper = v = S'GH}

= QS'G; etiam

COG = S'rG, _{ergo S'Gr « CGO Sc S'GC = rGO= 1 R.}

Sicque linea S'G, paraliaxin ftell« exponens, femper

tan-git circulum aberrationis,. eriam fi habetur motus reHuris ellipticus Sc variabdis. Hinc, _pofiro dimidio axeos

ma-joris in ellipfi parallaxeos = p Sc de cetero fervatis

iis-dem denominationibus ac in 11.III. _{1V., cum eft etiam} p(i - e2) cos v

S'r = S'G. cosv = , fit S'Q — b ßn _v

1 - ecosv '

p (1 - eA)cos v p{r-e2)ßnv

^ = A, Sc QSzzc b cos v : -bezzl,

t - e cos v _i -e cosv

fito nempe axe majore orbitae relluris in _{piano circuli} La-titudinis, _{quo fa£to AB congruir cum parallelo echprica?,}

punctum S transeunte. Si vero axis _{major orbitae}

tellu-lis non in _{piano circuli latirudinis jaceret, inclinario,} quam haberent _{parallell eclipticae,} ab, dz, pun£ta C Sc S

transeuntes* ad AB, esfet = Longit. ßellce - Longit. perih.s

) '39 (

quae differentia, ur antea,. ponatur =

d

(§.XL);

&

fi

S'q

ab S' ducatur ad de _{p.rpendicularis,}

_S'q

zzz A =

S'C.

ßn

S'Cp HH beßnd; eft vero

S'C . ßn S'Cp S'C .ßn {S'CB _- d)

= S'C . /w S'C£ ro/ d _-

S'C.

cos

S'C

B

.find

£

p(i - e2) foxy S'C.finS'CB== S'Q=:bßnv

&

i - ec osv pfi - £a)ßnv

S'C'cosS'CB = CQ z=.bcosv ^— ,

itaque

i - e cos v pfil - ea) cos v cosd

h zz b_{ßnv cos}d _4~

b

cosv

ßn

d

J i - e cos v p Ci - e2*) ßnv ßnd — 4- beßndfeu i - e cosir p {r - e2) K = bJfin_(v - rf) 4- .. cos

{v- d) -f-

be ßn

d;

i - ecos v

fi vero elongario folis a

conjun£tione

cum

ftella,

ut

fupra,

ponkur = E, erit v -

d

=:

E,

quoniam

elongatio

folis

ab _{apogeo eft v; ergo}

p (i - e*") cos E

Ä = b fin E H ~pr

+

be

ßn

rf..

.. i - * cox_fit -+•rf/

Porro _{quoniam eft Cc} zzz becos

d,

erit 5^= /=

S'C

cos

{S'CB - rf) - cos d;

eft

vero

S'C. cos[S'CB- rf) ₌S'C. _cos S'CB _{. cos}

d

S'C

ßn

S'CB.

find,

in _{qua ^xpreslione}

_fl

_{fubftiruantur}

valöres

ipforum

SC.c&s S'CB & S'C .ßnS'CBr fupra

alla ti

,

fit

) 30 (

p(l-tf®)finvcosd

ffiC.^ cos v(&CB- d)₌ b_cosvcosd- -f-bfin_vfin d i - e cos v

pj-e-) cos vfin d

-fr- , unde i-ecosv p{i -es) tzzz bcos_{(v-d) fin (v} - d) - becos d i ~ecosv P 0 ~ e<1) =£ cosE - _—r —-finE- becosd. s-e_cos£E-f-dy pQi - c*) b Pofitis vero , _

~{ETd)=r'

&7=iflWunde/»»--~ V(^ -+-

_r=y

~

₊ r>

j

C0SU

"

=

V(1

, crunt b cos E - r_{fin E} r= b cosec u _{fin («} - E) b_{fin E \ r cos E zu b cosec}

t1 cos_{(E - 1i) &}

t zzz bcosec u_{fin (u}

- E) - be cosd A=b cosec u cos _{QE - n) + befin d}

unde _{prodeunt formulae}

b_{fin (u - E) be cosd}

(I) /' =3 - &

finu cos Lat, cos Lat.

b cos _(E - u)fin Lat.

<TI) A'ss — _befin

d finLat.

finu j j

5. XV.

Calculus _{fuperior multo föret}

_fimplicior;

_fi

paralla.

)

c

xis haberetur conftans, quad in

determinandis

/,

a,

a','

os, errorem quantitate

o'/3oo84 minorem

adferrer,

fi

pa»-rallaxis tota f. axis major par.

eliipfeos

haberetur

Ut inveniatur, qninam tunc

föret

habitus

formularum

(I) öl (ii),

esfet

tantum p

in

iis

pro r

fiibftituendum;

valöres vero ipforum

l'

öl

a'

tunc

hoc

etiam

modo

fa-cillirne inveniuntur. Quoniam nempe

S'C

-4-p2)]

(fig. 2.) eft

conftans,

fit

via

apparens

ftellae

in

piano,

eclipticae

parallelo,

circulus,

qui

videtur

proje&us in

el»

lipfin, cujus axis

major

eft ad minorem

in

ratione

i:

finLat.;

öl _quoniam

_{efi: GCb}

_{= v -}

d

_»

E,

eric,

pofitis,

ut

an-tea, S'C =: g ÖL

S'CG [==

are

(tang

==:^)]

=5

a

(a) /' Qg cos

{E -f-

a)

-

be

cos<T)

sec

Lat.

ÖL

(/3) A' = (gßn

(E

-+-

fl)

-f-

befind)

fin Lat,,

unde patet

esfe

A'

maximum,

ubi

eft

E

=

90°

-

a,

po¬

fitis d öl latitudine

conftantibus,

&

=

o,

ubi

eft E

be_{fin d}

=- are {fin —

)

- a,

esfe

vero

/' maximum,

ubi

g

be cosd efi £1=* - a, öl =0,

ubi

eft Ezzz

are

(cos

=

—)

-

ß.

5. XVI.

Ut inveniantur variationes

declinationis

Sc

ascenfi-onis reifae, pofiro motu

teliuris

elliptico

Sc

variabili,

fit

S locus ftelide verus (fig. 4.), C centrum

circuli

aberratio-nis, CH, E'0 duo aequatoris,

CQ,

FR

d«o

eclipticae

pa-rallcli, punfta C öl S iranseuntes,

AR

portio

curvae

va-riationis, A locus ftellae apparens

in

eadem

pro

certo

) 3» (

SL, SK ad _{CQ, CH, quo fafto erit ST s= /, ATzz K,}

P (i - e2)

Se cc & Ae zz 8; _{ponantur eriam}

i-e cos_(E—4—$)

anguius pofitionis = P, anguius ACd zz C. Tunc erit

anguius HCd = OSR = P, Cd = / -4- CL tz / &£ro.r cf b cos E- r_{fin E & Ad=z (A}

- be.fin d)fin Lat. r= (£fin E

-f- r cos _{E) fin} Lat. (J. XIV.). Quatenus _{porro anguius}

Z/C5 ert _{anguius d proje£fcus, qui vocetur /,erit ex §. XI.} tång d! =zfin Lat. längd, & quoniam CL, in parallele eclipricae iitus, per proje&ionem non mutatur, ert idem

ac Cc (Hg. ₂₎ feu _{= be cos d, quare etiam erit CS(Hg. 4.)} zz be cos dset d & KS = becos dsec d'. _{fin^P+d'),unde}

$ = AC.fin (C _- P) •+• be_cosd_sec d'fin (P _-f- df_j Se

gezzAC. cos _{(C- P) -be cos dsecd\ cos (P -f- d'); ert vero}

Ad _{fin Lat. (bfin E-fr-r cosE)}

Tang C = — zz — t—— , unde

Cd b cosE - r_{fin E}

fin Lat. [b fin E-\-rcosE)

fin C zz

SOS C=

V[(ÄcosE-rfin E)2 +fin2Lat, (bfin E-4-r cosEfi~[

b cos E - r _{fin E}

^/\fibcosE-r finE)2 fin2Lat.(bfinE-\-r cosE)al

*

& inde

fin[C-P)zz fin C cos P - cos Cfin P

cos_{P(bfinE+ r corE)fin Lat. -{bcosE-rfinE) fin P}

~

Vt C6coSE-rfinE)2-^-fin2 Lat, \bfinE-±

rcosE)2J*

ert vero

AC zz _^/[(bcos

_E-rfin

_{E)2 +fin2Lat.{bfinE^r rcosE)2];}