208 MEMOIRS NATIONAL ACADEMY OF SCIENCES, VOL. X, NO. 7. 



TABLES OF THREE PLANETS.— ( IOI ) HELENA, (103) HERA, AND (119) ALTHAEA. 



The elements of these planets are osculating elements. 



The perturbations are developed in sines and cosines of (ig — i'g'). The form of develop- 

 ment is the same as for the foregoing group of eight planets, see page 202. 



Before tabulation, the expression for v was transformed into an expression for log 

 (1 +]/) = 8 log r by multiplying the expression for v by Mod. sin 1" and adding the higher powers 

 of v wherever they were appreciable. Similarly the expression for uj cos i was transformed 

 into an expression for <?/? by multiplying the expression for u/ cos i by a sin 1". « 



For tabulation the perturbations are segregated into their nonsecular and their secular 

 portions. 



NONSECULAR PORTION OF THE PERTURBATIONS. 



In accordance with the plan adopted by the original computers for tabulating the non- 

 secular perturbations of these planets, all terms having (ig — i'g') or multiples thereof for 

 particular values of i and %' as argument are tabulated under the single argument (ig — i'g') 

 for the three components in Tables II, III, and IV, respectively. The original numerical 

 designation of the arguments (ig — i'g') for particular values of i and i' has also been adhered 

 to, and differs for the three planets. The numerical designation of the different arguments 

 and the numbers added to make all quantities positive are given separately for each planet. 

 In Tables II, III, and IV the headings of the terms depending on a particular argument i are 

 (ndz)i, (d log r)i, and (<J/?) 4 -. 



SECULAR PORTION OF THE PERTURBATIONS. 



This is tabulated in the same manner as for the preceding group of eight planets, in Table 

 V, and is denoted by (ndz) t , (3 log r) t , and (dfj) t , respectively. The argument to be used for 

 each planet is indicated in the table. (ndz) t , (o log r) t , and (<??),. must be multiplied by T=t — t g 

 in Julian years. 



From the sum of the perturbations for each component taken from Tables II-V must be 

 subtracted the sum of the constants added to make all quantities positive. The constant 

 terms of the developments are not included in the tables and must also be applied. For each 

 component the algebraic sum of the constants to be applied is given separately with the tables 

 of each planet, and is designated by c z , c r , c„ for the three components, respectively. 



Besides the constants introduced to make all the values of d log r positive, and the constant 

 corresponding to the absolute term in the development of v, c r also contains the correction 

 which it is necessary to apply to d log r when the geocentric places are computed with the 

 value of the semimajor axis a which corresponds to the mean mean mption. It is to be observed 

 that in Hansen's theory the disturbed positions must be computed with constant elements. 

 The constant in the development of v corresponds to the value of the semimajor axis a, com- 

 puted from the osculating mean motion. The value a, however, given with the elements 

 and in the auxiliary quantities in these tables, corresponds to the mean mean motion. That 

 part of c r which is due to the introduction of the semimajor axis a, corresponding to the mean 

 mean motion in place of the osculating value of a, is indicated for each planet. 



Thus, in the tables for Althaea, page 348, the sum of the constants added to make 

 all numbers of Table III positive is 107.5 units of the fifth decimal place. The constant term 

 in v, page 350, is +27". 4, and the correction of this constant necessitated by the use of the 

 mean instead of the osculating value of the semimajor axis a is —38". 4. The algebraic sum 

 of this latter correction and of the constant in v is — 11".0, or — 11". sin 1" Mod. = —2.3 units 

 of the fifth decimal place in d log r. The total number of units of the fifth decimal place to 

 be subtracted from d log r is, therefore, 110 units = c T , as given on page 348, of the tables of 

 (119) Althaea. 



" It is to be observed, however, that the elements g and n contain the constant and the nontrigometrical secular 

 parts of ndz, respectively. 



