1G THE SATURNIAN SYSTEM. [§18-22 



. . . br — ibqcosy 



= _ b cos y log (/ 0(0) -J- Q — r cos (f) — . 



J 0(0) 



But this formula may not be used when o — r is also very small. 



18. The integration with reference to (p gives. 



R = E£ — B£ — B£' + B£', 



in which 



2© 



=1^' 



l 2 



' 



The formulae and notation of § 9 give 



2<3 2© 



&" = 



b fr--{-(z — b) a — (ircmcp . (z — S) 3 /* r 2 -j- (z — b)' 2 -Qrcosy 



', f r 2 +(s— b)* — Qr cos cp , (z — &) 3 C _^ 



r J$ fi ' '• Jd[r 2 +(;— i) 2 — j--cos-<jd]/ 2 







'^ "'J* L 2/, "i" i e r/ 2 1"8^r*/ i J"r: 8j/ 2 r J*Afl» 







2© • 







2© 2© 



I) 



2'P) 



z — b C q* -\- 2 q r cos <p 







g* ^— tov(=-ft) a E 



)• r — o 



. ^ - - y (*-*)' r v(f-co a H ^ + h. ^ 1 



_ r e 3 (3r +(.) 1 /»(« *»+«•/»+/& )] g — ft F i 



L '• ('• + e) "^ r («■ +/*>) J V C(e + »') 2 + («- 4) a ] v 



19. When the point is very near either of the edges of the ring, the forms of 

 reduction of § 10 give 



B£' = (b — 2) ^±^ [log (4 sec 4a) — 1 — tan i£ (* © — »£■)]• 



20. When the second dimensions of z and J can be neglected, the integral of E2' 



in § 17 gives 



2<3 2© 



J^ [5 cos y log (/ 0(0) -f Q - r cos 9 )] -J f __ qf «---_^ r - 



/* ftrsin X/oao+g) 

 JtfotoAfm + e— r cos qi) 







2'P) 



(278) 



