320 L EULERI OPERA POSTHUMA. Astron.mech 



totius sphaeroidis oriundae, quam carum momenta Yy^CY et Xx . CX; quae deinceps in unam vin 

 toti attractioni aequivalentem conjungi poterunt. Quo autem hae integrationes commodius absoh 

 possint, transformemus formulas v~^, w~"^» W~^ et {u)"^ in series, quae, si distantia centi 

 virium V(ff~^9g)> quam ponamus = ^, a centro sphaeroidis C fuerit valde magna, convergant. 

 Cum igitur sit v^Yihh — 2fx — 2gz -\- yy -^ xx -^ zz) ^ erit 



_3 1 Zfx-\-Zgz Syy — Sxx—Zzz 15 ffxx -^ 30 fgxz -^ i5 gg zz 



_3 1 Sfx^Sgz 3yy — Sxx — Szz I5ffxx — S0fgxz-^i5ggzz 



(") 



— 3 



A3 ^5 2^5 2A' 



1 Bftc-^Sgz 3yy — 3xx — Szz i5ffxx — 30fgxz-^i5ggzz 



h^ h^ 2A^ ■ 2A' ' 



/ \^s L 3fx — 3gz 3yy — 3xx — ozz i5ffxx-t-SOfgxz-^i5ggzz 



W "Jis ;^5 2^5 ' 2A^ ' 



Hinc igitur erit vis tota Yy ex attractione totius sphaeroidis orta 



T7- 8kkg /», , 7 /j 32/j/ — 3araj — 9zz i5 ffxx -i- i5 gg zz\ 

 Yy = -j^fdxdydz (^l 2^^^^ -H 2^;^ ), 



et vis tota Xx pro toto sphaeroide orta 



Deinde vero erunt momenta totalia 



Fy . CF= s fxxdxdydz -^ fxxzzdxdydz, 



Xx ,CX=^ fzzdxdydz-^^-^ fxxzzdxdydz^^^ ^^^ 



Qaoniam triphci integratione opus est, ponantur primo x el z constantes, ut obtineantur vires ex 

 elementis secundum rectas RM sitis oriunda, eritque 



Yr = ^-^frdxdz (i _ yy-^--^-^" ^ i^n^^J^m'y 



Xx = ^-^frdxdz (l _ !«-«^-3" ^ 15ff^H-^»5»i>» ^^ 



F^ . CY——j^fxxydxdz r~^ fxxzzydxdz, 



Xx»CX=—-~fzzydxdz -^ fxxzzydxdz. 



Concipiatur jam recta RM usque ad superficiem sphaeroidis producta, atque y determinar 

 debebit ex aequatione locali pro hac superficie sphaeroidica, inter coordinatas x, y et z expressa 



