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nes, on a tenu compte aussi bien du poids déterminé par la position de l'étoile sur la plaque, que des erreurs systématiques 1).

Afin de rendre les résultats des trois clichés directement comparables entre eux, on calcula ensuite les constantes à l'aide de 74 étoiles bien déterminées et communes à toutes les plaques.

Le tableau suivant donne pour ces 74 étoiles:

1o leurs coordonnées équatoriales pour 1900.0 d'après les déterminations du Catalogue

photographique du ciel;

2o les coordonnées rectilignes calculées pour le centre supposé
a=9"45" 2°.100; d % = 43° 0′ 13′′.100;

3o les coordonnées rectilignes mesurées sur les trois plaques 412 a, b et c, correction faite des erreurs de division du réseau.

Le numérotage des étoiles est le même que sur le cliché 412 original.

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1) Pour ces deux données, voir ma thèse: Sur la précision des déterminations photographiques des po

sitions des étoiles. Helsingfors 1906.

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39 9 43 37.940 42° 54′ 40.75

38.949 42 44 19.29

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+ 10.8744

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+60.9694

+ 41.4350

- 15.8929

- 19.0671 - 20.0652

mm

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- 59.0164

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- 13.6103 - 18.3108

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8.207750.5709

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5.746130.5815

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3.7182 - 15.6041

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3.7120 - 19.8761

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4.0792 15.6787 1.6347

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- 54.2435 + 9.9842 - 54.2236 46 12.520 43 56 9.74 +12.7056 + 55.9731 +13.2922 +56.3853 + 13.1496 + 56.3026 + 12.9361 + 56.3168 12.236 42 48 5.34 +12.8916

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6.427 30 13.05 + 11.8810 24.368 43 32 17.34 +14.9406 32.312 42 17 31.07 +16.7188 47 15.857 43 48 43.24 + 24.1827 16.062 7 6.0124.4969 9.444 42 35 50.16+ 23.4845 + 23.8807 17.219 33 19.82 + 24.9356 - 26.8041 + 25.4921 26.5691 + 25.3328 32.096 43 57 20.05 +27.0545 +57.2250 + 27.6398 +57.6324 + 27.5064 35.978 16 28.07 +28.0673 +16.3586 +28.6414+ 16.6853 +28.4904 18.565 42 57 41.04+ 25.0184

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+ 45.4056 +34.9829 + 45.3133 + 7.2225 + 32.7723 + 7.5237 +32.6304 + 7.4421 48 34.323 55 15.16+38.3008 +55.2451 +38.8981 +55.6592 + 38.7535+ 55.5572 14.860 30 5.03 +35.0290 + 30.0367 +35.6157 +30.3912 + 35.4614 - 21.5054 + 38.6952

+45.0251 + 35.1232

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- 21.7283+ 38.8509

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Pour les trois clichés, on établit des équations de condition de la forme

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Avant que ceci ait eu lieu, et afin de rendre les quantités net n aussi commodes que possible pour les calculs, on fit sur les x et les y mesurés des corrections du centre des grandeurs suivantes:

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Toutes les ordonnées furent de plus corrigées de la partie principale de la valeur de l'échelle -0.002 1). On choisit comme unités 10' pour x et y, et O'.01 pour n et n,. On alla dans les calculs jusqu'à un dixième des unités employées. La première équation de condition pour x du cliché 412 a par exemple se présente ainsi sous l'aspect suivant:

a -6.3b+1.3 c + 2.8=0

Les équations normales formées par la méthode des moindres carrés furent alors pour 412 a:

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singfors.

en y:

74 a' 28.10 b'+ 33.20 c'+ 64.40=0

754.82 b'

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10.98 c' 630.07=0 844.44 c'+166.92 = 0

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1) Voir à ce propos l'introduction du t. IV du Catalogue photographique de l'Observatoire de Hel

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Le cliché 412 c, enfin, donna le système suivant d'équations normales:

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Un coup d'œil sur les coefficients des équations normales et des équations d'élimination montre combien elles se rapprochent pour les trois plaques. Les erreurs des positions des étoiles de répère doivent par conséquent influer sur la détermination des constantes par des grandeurs presque exactement identiques pour les trois clichés. Introduisons maintenant les corrections de la réfraction pour b, c, b', c', afin de rendre directement comparables entre eux les résultats relatifs aux trois clichés. Nous réunirons alors les corrections de l'aberration et ceux de la réfraction, comme cela a été fait dans les tomes du Catalogue photographique du ciel publiés par l'Observatoire de Helsingfors. Ces corrections auront les valeurs suivantes:

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Si maintenant nous corrigeons b, c, b', c' des grandeurs mentionnées ci-dessus, on trouve, pour P, Py, "a ry, ainsi que pour leurs différences P-P, et r-r,, les valeurs indiquées dans le tableau suivant:

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r, x1

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Ces quantités sont exprimées dans l'unité employée d'ordinaire dans le Catalogue

photographique du ciel, c'est-à-dire 1'.

Les chiffres de ce tableau sont intéressants à un double point de vue. D'abord ils prouvent ce qui a déjà été constaté pour les clichés de l'Observatoire de Helsingfors, c'est-àdire que la valeur de l'échelle reste la même pendant la même soirée d'observation. De plus, ils montrent que P-P, aussi bien que r-r, ont à peu près les mêmes valeurs pour les trois clichés. En admettant que P En admettant que P-P, et r-r, devront être égaux pour les trois clichés, on peut calculer à l'aide les nombres ci-dessus les erreurs probables pour ces différences. On trouve alors:

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De ces résultats, on peut tirer à leur tour les erreurs probables des déterminations pour ainsi dire photographiques de p et de respectivement, en supposant R (p) = R(p) et R (r) = R(r), égalités qui doivent avoir lieu si l'on n'a égard qu'aux erreurs photographiques des positions. On obtient

R(p)=R (p1)=0.0000055

R(r) = R(r) =±0.0000177

Comparons maintenant les constantes calculées ci-dessus avec celles qui résultent du cliché originaire 412, en faisant usage des 74 mêmes étoiles que dans les calculs précédents. Je ne reproduirai point ici les résultats des mesures relatives à ce cliché, pris le 21 mars 1893; ils sont publiés déjà dans le tome IV du Catalogue photographique du ciel. Je n'indiquerai donc simplement que les équations normales et les résultats des calculs 1). Equations normales:

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1) Je laisserai de côté dans la suite les coefficients de a', b', c', qui sont égaux aux coefficients de a, b, c.

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