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



By equations Z (43), (235), (236) and the equations preceding (233), the factor tj and the 

 arguments J, £„ di are given in equations (238) in terms of osculating values and functions of 

 perturbations, inclusive of first order. 



To these should be added 



and 



3/ 2 \ 1 

 ri = r + ^l 1 -gTjo cos e^jiy sin s^-z cos £o) + 2')o2+ 



where i 



A=2'i + '^. + ^i 



•T =2^0 + ^0 + ^0 



The equations (233), (235), (236), and (238) permit the construction of two tables which 

 determine w, nor a, and e and tc. From here on the developments differ in form from v. Zeipel's 

 although they are the same in principle. If v, Zeipel's equations (237) and (239) are used, the 

 term {^2" — iiy^") should read 



(V + X3" + x,") - ij (1/2" + 2/3" + y.") 



in agreement with Z 91, line 14. 



Suppose that w-w^ has been computed by equation (233) and the argument F has been 

 introduced. The arguments and factors are unknown. 



By laylor 3 theorem 



w-w,=firi„ r, d„ j„ ^<,)+^/rj,+^jr, + ^Jd,+-l^n,+^^Ji„+ .... 



Inclusive of second order in m', the differentiation is for first order terms. 



Substituting the values of Jij, JF, Jdo, i^o. ^^0 from equations (238) and the additional 



equations above, 



/ ?i -f ^ -f ^•f\ 'I / ^ -f ^"f ^^\^ 

 w-w,=f{r)„ F, d„ Jo, ^o) + (^(2^-^^-^j4^2+(^^^+^ + 0-J2'?„3 





2(1 -JJo cos ^o'l^ + ^f 1 -fjlo COS ^oj^jCy 



sin £„— 2 cos £„) + • • 



The order of calculation is: computation of equation (233), in which the arguments and 

 the factors are given the subscript zero, differentiation of first order terms, computation of the 

 second order terms in the above equation, and the additon of these second order terms to the 

 first calculation. 



With some foresight the computation can be simplified. The arguments should be arranged 

 in groups like the following: — n7~' + 2^ + 2J 



-(r!,-l)r + 2^ + 2J 

 -(n-2)r + 2^ + 2J 



(7i,-l)r + 2/? + 2J 



Then, for whole groups of arguments, 



Sf Sf df_ 

 5(2^o) 5J„ 81\ 



Also for some particular argument in a group, the condition 



dJo^SF^S^o 

 may be satisfied. 



