Researches on the Distril.aition of the ÄTean INIotions of the Asteroids. 



I obtain 



da 

 dio 



= Wa y^e sinhv + cos wll + -^ cos w j fl + e cos iv \ 

 É^= W V^ui-iD + cos ni 1 + -|- cos uAi cos w + -^ ^iirw J 



-= W —sill ic cos /Ü + sill 10 cos yrf 1 +^cos }c\( 1 — ö"^*^* '^^j 





— — = —2 IF Sill tüfl — f cos «V + -;r- -^— 



dw 2 rf/r 



Or neglecting e'^'^ and simplifying 

 TF=-w«-W-v(l--|cos»„.)'^'{l-(.2 + ^-,-Jj^j,^p|^>cos,.} 



^^* TFa(cos?r + 2e-Apcos"';tO 



(8) 



dw 



-= ^7(1 + 4- cos w — ^ cos-^z^;) 

 ; ^ 2 4 '^ 



e— — = — W -r sin w cos-iv 

 div 4 



fZS OTT7/ ■ • , , 6' dv3 



— — = — 2lr (sm lü—e siii ic cos ?r) + -T; — ^ — 

 diu 2 rtR' 



6. Tu find the secular variations of the elements in a unit 

 interval of time we have to evaluate 



r da ~l _ ju^ r '^"^ da -, 

 V. dt A '2,nJ(\ dw 



etc. 



etc. 



Since the effect of the resistance is supposed to he very small, we 

 may neglect cVo^ within a single revolution, so tliat the quantities 

 n, a and e may be integrated as constants. 



Now if we put 



X—{\—q cos-zü)"2"siii'?r cos'/t^ 



where (/< 1 and .^, /, j are any positive integers, it may be easily 

 seen that 



