9 



12-17] THE SATURNIAN SYSTEM. 15 



12. When the second dimensions of z and b can be rejected, the preceding formula 

 becomes 



Q? = - I2J" = 2 b (<? + r) EL + 2 b («> - r) F} M . 



13. When the attracted point is so near the axis that the square of r can be re- 

 jected, the integral of the formula of § 7 gives 



ill' = — 2 'O «/ log (/P + * — *) — 2 © (* — W- 



14. The attraction of the ring in the direction of the radius r, and toward the 



kR = — D r S2 = k 



>• Q COS (J) 



~V 3 " 



15. The integration, indicated in the preceding formula, may be first performed 

 with reference to L, and the result is 



in which 



1*2= p—f • 



J 0(0) J 2 



1G. The integration with reference to Q gives 



D,B = B£ — B£ — R£ + B£, 



in which 



K =f 9 (V %)■ 



The notation and equations of § 5 give 



r p — o 2 cos if = — p 2 cos <p — p r cos 2 <jp -|- r 3 cos cp sin 2 <jp 



= — / 2 (0) cos (/i — j» r cos 2 y -)- 2 r 2 cos <f sin 2 tp, 



B :=(.-») J j;^+(.-l)r«.S t X 3 £j l 



— 2 (2 — 5) r 2 cos ai sin 2 9 / -^ — 7. 



Jp ./0(0)/2 



= (2 — b) cos 9 log (/ 2 -f- — r cos <p) — r cos 2 <p log (/ 2 -4-g — b) 

 — rsin2g>taii [ -i££=?. 



7 r sin qo/ 2 



17. When the second dimensions of £ and 5 can be rejected because they are 

 small, the value of R 2 can be reduced to 



„, hr — Spcosqp 



1»2 — ft ~ • 



J 0(0) 



This gives 



R 2 ' = — /; cos w j -j. b r cos 2 y / -| — (- 2 J r 2 cos 9 sin 2 </> / -5— 



(277, 



