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



Certain other transformations of the elements which v. Zeipel makes are omitted. Those 

 terms of the perturbatioas which have the argument £ have the same period as the planet and 

 can, therefore, be absorbed in the elements. It would be necessary to set up formulae for this 

 transformation to mean elements, and it is not profitable to do so. 



PERTURBATIONS OF THE RADIUS VECTOR. 



The perturbations in the radius vector are computed in a manner similar to that for 

 {nhz — [n^z\). In Table XLIII the numerical coefficients are multiplied by their respective 

 factors w*, jjp, r)'i, j^, the terms are collected, the kno\vn parts of the arguments are incorporated 

 in the coefficients, and the terms are grouped in the form: 



v= la sinx + -^& COSX+- • • • 



+ (!?-!?„) {Ja' sinx+^6' cosx+- • • •} (35) 



+ (!?-.>„)^{2'a"sinx + 2'6"cosx+- • • •} + • • • • 

 Let 



o = — ^ sin K b =Jc cos K 



a' = k' cos K' ¥ =¥ sin K' (36) 



0"= -V sin K" h" = l-" cos K" 



Then 



v^Ikco3(x+I0 + (^- ^0)^^' sin (x + K') + (t? - «?„)='i'/<.-" cos (x + A"') + • • • • (37) 



and to correct the perturbation for the use of the improved value of the mass, v should be 

 multiphed by 1.00050. 



If the mean motion n^ is adopted, the constant in v must be corrected by 



2 n^-n, _1 



3 rij sm 1 



This correction of the constant in v permits the use of the relation 



n2V/ = P 



in the computation of a geocentric place; without this correction it would be necessary to use 

 the relation 



in the determination of the parameter p. In the computation of the eccentric anomaly it 

 is permissible to use either ?ij or tIj, for the difference is taken up in the modification of ndz, 

 but the theory of Hansen demands the use of constant elements. Hence, strictly speaking, 

 71, must be used in computing a geocentric place. The modification of the constant in v renders 

 the employment of n^ equivalent to the use of ti,. 



(10) Hygiea. 



2 71,-71., 1 2 054067 1 _ ^^,0 



3 71, sin 1" 3 637.''3 sin 1" ' ''^ 



The constant in Table XLIII or Cis +47f6. Therefore, the new constant is: 



+ 47.'6 - 87.^8 =-40f2 = (1.604) cos 1S0?00 



where the coefficient is logarithmic in seconds of arc. 

 The perturbation is tabulated on page 27. 



