14 



MEMOIRS NATIONAL ACADEMY OF SCIENCES. 



[Vol. XIV. 



To make the order of computation evident, the successive steps for a group of terms for 

 Hygiea are given. 



The second column contains the sum of the niunerical coefficients multiphed by their respec- 

 tive factors w'rjP-q'i'f-'. The columns —S and + C contain the required terms from this group in 

 the table. They can be computed at the same time if a traverse table is used.' 



From S and C the elements tz and ip can be computed by the formulae: 



e sin {it — 7ta)==S cos v'u 



e cos (;: — ;:„) =^0 + '^cos Vo (H) 



e = sin <p 



In place of -q^, J^, !„ the following are used hereafter: 



(12) 



There remains one more element to determine, namely, c, but the computation must be 



deferred until vs'e know the perturbation ndz at < = 0. (See equation (1), page 10 or page 16.) 



The fictitiously disturbed eccentric anomaly at the time t = denoted by £„ is determined 



through the relations : 



£o-eo sin fo = C(, (13) 



where £„ is calculated with the aid of Astrand's table ' ; 



ifi'Kvo-To) =y J-^" fgho> 'i'i^i = -\/r+i ^si^'^o-^) 



(14) 



£o=131?3236 



(10) Hygiea. 



r„ = 3?2968 



£, = 127?6064 



£,-esinf, = 122?5578 



The perturbation ndz i.s computed as follows: 



The function 1 +<i!>(t?) is computed from Table XXXVIII or B. The coefficients are mul- 

 tiplied by their respective factors, the trigonometric functions of the arguments are expanded, 



and the coefficients of . * ii? are collected, (j is the numerical coefficient of t>). 

 sm 



' Memoirs of the National Academy of Sciences, Vol. X, Seventh Memoir, p. 218. 

 'Hulfstafcln zur leichten und genauen AuflSsung des Kepler'schen Problems (Leipiig, 1890). 



