16 ]MEMOIRS NATIONAL ACADEMY OF SCIENCES. [Voi.xiv. 



Unit of A^, etc., is one radian 



[7i&].= (3.59592) sin 2C +|(C-Co) [(0.933„) sin L'c 

 + (4.09785) cos 2^ + (0.521) cos 2^ 



+ (3.0783) sin4C +(0.085) sin 4C 



+ (3.2230„) cos 4C + (0.005) cos 4^ 



+ (2.4390„) sin 6C + ] 



+ (1.494„) cos ec + 



+ 



in which the coefficients are logarithmic in seconds of arc. For this planet it is not necessary 



to include C\". 



In equation (IG) let 



iS„ = Ic cos K (7„ =^ sin Z .^_> 



5'„ = - it' sin Z' C'n = ¥ cos K' ^ ' 



Then 



[ndz\^nc sin (nc+ii0+ |(C-Co)^^'' cos {nC+K')+ (18) 



The argument ^ is given by the relation : 



and ^0 is the value ol t^ &t t = Q, in which, [n'dz'], the long period term between Jupiter and 



Saturn is : 



[n'dz'] = (9.5215) sin{ (9.58539) T+ 1 15?32C} (20) 



where the numerical coefficients are logarithmic in degrees, and T is measured from the date of 

 osculation in Julian years. 



The complete expression for the long period term in ndz is : 



[ndz] = [n8zl^^^^j^^^lls-in'3z']) (21) 



It is important to remark that, in equations (19), (21), the eccentric anomaly is computed 



by the usual formula, 



s — e sin s = c + nt + ndz (1 ) 



m which the multiples of 2;r must be retained, for s is used here as if it were the time. Since 

 ndz is unknown, the computation is by successive approximations. 



UO) Hygiea. 



M2],= (4.11837) sin (2^+ 72?5246) 

 + (3.3130) sin (4^ + 305. 627) 

 + (2.442) sin (6^+186.48) 



+ 



+ |(C- Co) {(0.963) cos (2C+ 68.83) 



+ (0.199) cos (4c + 309.75)+ ....}+.... 

 in which the coefficients are logarithmic in seconds of arc. 



The argimient ^ in (ndz -[ndz]), the short period part of ndz, is given by 



^='L^[nd2], + c (22) 



and the function itself is computed from Table XXXV or A. 



