30 



of thé earth in this system, .r -|- §, y + ^i, ^ + ? those of the moon, 

 then the projections of the pertnrhative force on the three axes 

 are given by the expres.sions : 



dV\ dV /^dV\ dV /'dV\ dV 



The ratio of the distances sun-earth and earth-moon being very 

 hii'ge, 1 develop) in powers of |, ii, 5, neglecting second and higher 

 powers. Then the expressions for the perturbative forces are: 

 d'V d'V d'V d'V d'V d'-' F d'V d'V d^V 



o.v^ divoy oxoz dxdy Oy^ dydr dxdz óyóz oz' 



and one can inti'0(hice as the pertnrhative function the function 



R 



d'V d-V d'V d"-V d^F d^V' 



dx'^ dy' ^ dz'^ dxdy dxdz ^y^z_ 



Here for .t, y, z are to be substitnted their expressions in elliptic 

 elements and then the secular portion of R is to be taken. Since 

 the powers and products of ;;, ij, C, contain oidy the elements oftlie 

 orbit of tiie moon, the coefficients on the contrary only the elements 

 of the orbit of the earth we can take the secular portion of each 

 separately and multiply these together. 



Besides the system jnst mentioned suppose another system .i;', //', 2', 

 the sun also being at the origin, but the axis of z perpendicular to 

 the equatorial plane of the ellipsoid. Then w^e have 



z = X sin *I) sin J ^ — y cos <P sin ,/„ -j- z cos J^^ 

 therefore 



— - ^ sni si?i Jp ;. -^ = — cos<P sinj^; -— = cos J^. 

 Ox oy Oz 



Perturbations caused by the jirst ellipsoid 



V 



From the expression given for i2 == ^ we deduce, neglect- 



k'^ :rr qa^ c 



ing the terms having sitï' J as a factor : 



ö'<2 r du Ax^ 



r du 



bx^ J {a^+uYic^^iifk {a'^-Xfic^^/.f-i 



Ö' « _ 4 X y 



d'i^ Axz' ^ r* du 



— {a^—c'')— 2 Ue — c^) sin sin ./„ - 



