NO. 3.] MINOR PLANETS— LEUSCHNER, GLANCY, LEVY. 141 



Therefore, eq. (194) becomes 



Comparing the coefficients of like powers of w on either side of the equation, it is evident 

 that the integial must be of the form 



S =S-i^^ + S, + S,w + S,u^ + 



Substituting this relation in the preceding equation and equating like powers of w, the system 

 of equations (IGS-,) — (195,) follows. 



Within the extent of the following developments one more equation should be written by 



analogy. 



dW 

 This system of equations is integrated in a manner similar to that for -j- (see p. 81). Each 



equation is broken up into two equations, one a function of s and one independent of e. The 

 differential equation (194) is then replaced by eight differential equations, the integrals of which 

 can be obtained in the order, 



S_., (5,-[S„]),[5o], (5,-[SJ),[,SJ, 



dW 

 As in the case of -j— , the condition is imposed that 



The equivalent equations are (196)-('200). 



dW," 

 A comparison of the differential equations for (-Sj — [^S",]) with the expressions for — t^ 



dW " 



— Tj— leads to an analogous form of integiation for certain terms. Within the extent of our 



developments, 

 and 



D 



-i (1 -e cos s) ^J{I,-[I,])d,-[(l -e cos e)^JiI,-[I,])de] 



bW" bW" 

 take the place of -. ' and —^~—, respectively. Without change of notation for the third 



coordinate, (iS — [S]) is given by eqs. (201), (202), where F, G, a are computed from F, G, H in 



Tables XII-XTV', by means of eqs. (118) and (119). The coefficients V', G, a are tabulated in 



Tables L to LII. 



The function [S] is obtained from the integration of eq. (203). A constant of integration 



is added, which is the same in form as Hansen's constant of integration for the perturbation of 



the third coordinate, namely, 



Ci(cos i/- — f) +f2 sin 4> Z eq. (204) 



where c, and Cj are undetermined. 



It 

 By eqs. (192), the pertubation ■ is derived from 



" -. = S 



I cos % 



The perturbation comprises the computed value of eq. (202), the trigonometric sine series given 

 by Tables L to LII (which can be written by inspection with the aid of Table XV6), the series 

 forming Table LIU, and the constant of integration (204), in all of which 



