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France, 1, 313. D ' A l e m b e r t , J . (1754). Recherches sur différents p o i n t s i m p o r t a n t s d u S y s t è m e d u Monde, P t . 1. " T h é o r i e de la L u n e " . P a r i s . D e l a u n a y , C. E . (1860). Mém. Acad. , Paris, 2 8 ; (1867) ibid. 29. D e Sitter, W . (1917). Mon. Not. Roy. Astr. Soc. 77, 155. E u l e r , L. (1753). " T h e o r i a M o t u s L u n a e exhibens o m n e s eius i n a e q u a l i t a t e s " . Petropolis. Euler, L. (1772). " T h e o r i a M o t u m L u n a e n o v a m e t h o d o p e r t r a c t a t a .
64) where Δ = 1 —sin In! 04). 67) — sin/3 cosA sinjS sinA cos/3 53 2. LIBRATION OF THE MOON In order to compute dX/df and dßjdf we have to find the derivatives of the physical libration r, p and σ relative to / . 68) · sin/^. 63) having on their lefthand side the fundamental unknowns of the problem and, on the righthand side, the independent quantities SQ—sc, permits the carrying out of the adjustment in accordance with the principles of the least-squares method. This process, applied to the Dorpat heliometric series of Hartwig, gave the unknowns of the problem with smaller mean errors than those obtained by alternative ways of adjustment, and rendered it possible to obtain a strict value of the mean error of one observation (Koziel 1949a7), which could not be obtained by Bessel's method.
LIBRATION OF THE MOON libration of the Moon : | miWsinAo -cosA 0 \ x cos/30 . AC) U 2 AcosA 0 sinA0 / Δβ Δλ = where {wi} = Λ 2 . Δ . As, M Λι, Λ 2 and Λ3 denoting the the cracovian Λ = r(L), L, 0 sin ^So . 10) ) = Λι . Δ . Λ 3 , fas} = Λι . Δ . 11) first, the second and the third column of and — Δα — cos αο . 12) ! ΔδΙ. 12) cos ao . Δδ sin ao . Δδ 0 ) Formulae (2-10), containing rigorous analytical values of the first derivatives as coefficients at the corresponding differentials, may be used with advantage when the cracovian Λ or L is already known from other lunar calculations.
