31 



ö'<2 _ /• du 4y* 



d'£2 4V2' , r du 



^ , = : — '- ~r V' — ^■") + ^ (<^' —c^) <^os sf// J„ I 



a.vOc («^ + ^r (o' + ^-Y'-^ J Or + »)" (r' -f ^tyu 



d^£i r du r" dn ■ 



Substituting the elements of tlie oiltit of tlie eartli for .r, //, 2 and 

 neglecting the second and highei- power of the excentricity I get : 



-^ = - 2 C , + __ = . =^ O ; 



dx^ a^-p Oir Owdy 



d"-S2 2{a—c') 



dn *I> sin J ^ — 2 {a' — c^) i\ sin <I> sin J„ 



d.vdz «i^p' 



cos <I» sin J„ + 2{a^ — c^) Cjios sin ,/, 



= - 2C, - 2(a''-c^) C3 



02: ■' 



Let o be the radius vector, v the true anomah', m the longitude 

 of the perigee, cTl the longitude of the node, / the inclination of the 

 orbit of the nioon, then we have : 



5 = () [cos (?; -|- to — J^ ) cos '_'l — sin{v-\-co — j^, ) si n Sh cos i] 

 Tj z=: () [cos (^) + tu — S\)''^^'^Sl) -\- sin{v-i'<Ji) — S}i)cos^cosi\ 

 ^z=zQsi?i{v-\-öi — • j^) sin i . 



I write these expressions thus: 



Q {A COS V -f- -^ si7i v) 



11 ^ f) (C COS V --(- J> sin v) 



Q := Q [E cos V -j- 7'^sm ?'), 



A, B, C, D, E, F being expressions not containing the trne anomaly. 

 F'or the formation of the required products we need the secnlar 

 portion of (f co,s^ v and q'^ siif v; I get: 



S(,^cos^v=za\"[h -f 2<;^) S i^'' .nir v = I, a ^' {\ — e'') 



a\ being the semi-major axis of the luiuir orbit. 

 Thus we get expressions as : 



