12 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [Voi.xiv. 



Jupiter. 

 Epoch, 1851, Sept. 17.0, Ber. M. T. 



ME.A.N ELEMENTS. 



n'= 299fl284= 0?0S30912 

 ip'= 2°45.'95= 2?7658 

 Tz'= 11 54.45= 11.9075 

 ft'= 98 55.97= 98.9328 

 co' = 272 58.48 = 272.9747 

 i'= 1 18.70= 1.3117 

 Co = 126 59.81 = 126.9968 c'=199 57.70=199.9617 



Mean equinox and ecliptic, 1850.0. 

 Epoch, 1851, Sept. 16.96279 Gr. M. T. 



The following notes in regard to these elements are of importance: 



Jupiter's elements were first taken from Z 139. They were used only in the equations 

 nimibered (1). In these equations either set of elements may be used with sufficient accuracy. 

 In fact, it is not necessary to know Jupiter's elements as accurately as those of the planet, for 

 they appear only in the arguments of the perturbations. We have adopted Hill's values of 

 the elements and Newcomb's value of the mass of Jupiter. The tables of the perturbations 

 are based, however, on Bessel's value for m'. To correct the perturbations for the improved 

 value, it is only necessary to multiply them by 1.0005, and this is done in the formulae which 

 follow. 



The original epoch of Jupiter's elements was 1850.0 Gr. M. T. It was changed by the 



formula c' = 148° 1.'97 +n'< (4) 



The elements of Hygiea are very good osculating elements, computed by Zech. They 

 include perturbations by Jupiter, Saturn, and Mars and are based on five oppositions. The 

 reference for these elements is doubtful, for in Astronomische Nachrichten 39, 347, the elements 

 given by Zech are not identically the same, although the differences are very small. The values 

 given by v. Zeipel were probably taken from Zech's manuscript, to which he had access. They 

 may, therefore, contain some later corrections. 



The auxiliary quantities *, *, J are first computed by the formulae : 

 sin ^ J sin ^ (* + 'l>)=sin -^ (Q.o- SI') sin ^ (ig+i') 



sin 2 J cos 2 ('!' + *) =cos 2 (.Sl^-^') sin ^ (%-*') 



cos 2 "/sin 2 ('*'-*) =sin ^ iQ,o-Q,') cos -^ do+i') (5) 



cos 2 e/ cos 2 ('*'-*) =cos 2 (9,0- ^') t'os 2 (io-i') 



rv, 1. sin^ _ sin'l> sinfSJp— S^') 

 ^'^^'^^'- sini„~sini'~ sin J 

 Then follow 



n. = ^„-ft„-*; Vo = J; n'=;r'-fi'-M'; ,'=|'- 



y2=sin^ '2 cos^ k 'Po cos^ o ^'' ''='*^" "^ ^'^'^^ 2'''' ^^■^ 



j„=n„-n'; j„ = n„+n' 



Wo=^— 



