25 



for a point outside the ellipsoid a is the positive root of the equation 



00 



1 — 



; for a point inside I is zero. 



i;" -[- y^ -|- ^- = r^ we 

 z'^{a^ — c') \ du 



a--^X c^ + l 

 Putting V=:z k-Jtqa-c^ and .«"-[- y' -[- ^- =: /■' we have 



-Hu (a^ + z^)(.« + u)y {a' + u)y/c''-^u 



1 '^ i- = 0. 



Perturbations caused by the jirst ellipsoid. 



I develop in powers of c' = ?, S being a small quantity ; for that 

 purpose we need (neglecting terms of the third order) : 



r^2 — a- 



(' 



+ m)'/2 



d'i^A (a'— 0==)= 



I put >' = a, (1 + ^) and develop the part of ii independent ot 

 5 besides the coefficients of the different powers of ^ in powers of §• 

 Introducing the quantities : 



^ ^ r[ ^^^^ ^ ^ r°° ^^ (. ^ r! 1"_ 



p 



we get: 



§' / 2 2 \§» 

 i2=: C, -a/C, - 2 a^H\_ ^ + (2 -^VC,) - + (^ - - - - yj ^ H- 



\^2 3 2 y /) «1 p ( 



/ 27 15 35 \ 1 



+ (4 -}- 5y + 5y-^)g^ + (^- 5 - - ;, - - y^ _ - / J §^ j + 



"^ 2 a,'p' 



Let V be the true anomaly of the planet, if? the angular distance 

 between the ascending node of the equatorial plane of the ellipsoid 

 on the orbital plane of the planet and the perihelion of the orbit, 



