78 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [Voi.xrp. 



INTEGRATION OF THE DIFFERENTIAL EQUATION FOR TF. 



With the exception of Tables LVI and LVII all the following tables are concerned with 

 the integration of functions whose coefficients can be derived, more or less directly, from the 

 preceding tables. The terms of first order in the mass, before and after integration, are of the 



^Cp.,(n+r.-n + s),P,v{cii 



A+s-tJ) 

 A-e + (p 



where Cp.g = C„.p., + C,.p.,- w + C^.p.g-uj' -\ (see Z 25) 



and A = [n + r-i{n-s)]£+{n-s)d+in+i' W 



In the argument A the factor n is always a positive integer; the factors r, s, i, and i' are 

 positive and negative integers. Evidently, the factor of s is i-^ where Jc is any positive integer, 



and the arguments in a series are I^^r-s-^- Within the extent of Bohhn's tables all of the 

 coefficients can be written in symbolic form from B 188, XVII, XVIII. In the notation for 

 the coefficients the particular values of r and s are given, and there remains to be foimd only 



the positive value of n, if there is one, for each multiple of -^ • 



The following tables present, in skeleton form, any series of the given type. There are 

 properly two tables, one for perturbations in the plane of the orbit, and the other for perturba- 

 tions perpendicular to the same. The headings J and I are defined by 



j=n-n' 

 i' = n + n' 



Considering first the tables referring to the plane of the orbit, omitting for the moment 



the arguments bearing the subscripts ±5 or ±a, the argument A for any term is read from a 



Ice 

 main heading ± -^ and the first two columns under this heading. The tabulated numbers are 



the respective factors of d, J, and I. The degree of the factors in the eccentricities is indicated 

 in the subscripts p-qin the symbol for the coefficient. Further, when j^ = 1 



i n+i'n'=n(n-n')=ni. 



Hence the coefficient of i is also the number n in the proper table of the numerical values of 

 the coefficients. For instance, in the function T^ (Z 41, eq. 82) we have for one term 



F,.o(n-i.-n));sin (e + 40 + 4J) 



where F, taken from Table VIII, is numerically 



F,.„(n- 1. -Ti)„^= - 1514" +5780"w-8976"'U^. 



Adding e - </> to the argument and taking the coefficients from Table IX, we have also in the 



function T, G,.,{n-l.-n)r,^^T) sin {2e-4> + 4d + 'iJ) 



where <?i.o(w- l.-«)«=4= +452"- 1475"w+ 1451' V. 



In this manner the series is built up. 



The coefficients having subscripts ±d and ±a belong to terms depending upon the mutual 

 inclination of the orbit planes. They differ from the preceding type of terms in three ways. 

 In the fiirst place the subscript signifies the addition of ± J and ±2 to the argimient, respec- 

 tively. Evidently, if ±i is added to the argument, the factor of J is not n but7i±l, from 

 which we determine n. Lastly, these terms contain the factor f, i. e., within the extent of 

 our tables the exponent t is not greater than unity. 



For the tables referring to functions which concern the perturbations in the third coordi- 

 nate the same explanations hold, with the exception that the additional subscript ±7t' signifies 

 the addition of ±11' to the argument. 



These tables, in connection with the proper tables of numerical coefficients, enable the 

 computer to write a complete series by inspection or segregate any term of given degree and 

 given argument. 



