18 THE SATURNIAN SYSTEM. [§26-28 



in which 



2<3 2© 2© 



7'" _ /-. i\2 C r*+(z—by — Q rc os(p . , C f \1 I ,fl p. \ f 1 







=..2 y/ (* - i ) 2 [^rE^ ( H , tf - * ®) + H.", ^ - * ®] 



26. When the ring is thin and the attracted point near its plane, so that the third 

 dimensions of z and b can be neglected, the value of Z 2 ' can be reduced by the formula) 



f—f 4- (z ^ br 



log (A + Q — r cos (/) ) = log (/ 0(0) + <>-,■ cos 9 ) + ^ (y^+^I — y 



The substitution of these values in Z 2 ' and the neglect of the terms which are common 

 to Z 22 and Z^, as well as to Zj 2 and Z n , gives 



„// — s b zbr cos <-/> 



/oiO) /o(0) (/o(0) + e — ?'COS <J>) 



—zb ~ & <*os <y (/o ( (i) — g + r cos <y) 



_sb (q cos <p — r) z J cos qp 



r/ 0(0) sin 2 <j) rsin 2 g> 



J /(> — r (' ~r r \ J o cos <p 



0(0) \siuHg> cos' 2 1 <jp/ r sin 2 qp " 



The integral of this expression becomes, after omitting the last term, for reasons already 

 given, 



71" ~ zh f \- 1 (°~ r e + r )1 



2 " r (q -{->') J 6 lV (1 — sin 2 2* sin 2 gi) Veos 2 qj sin 2 rp/J 

 o 



2b, p JIjl™ 



27. The mass of the ring is 



& 



which aives 



s 1 



M 



2 /Q ^(al—a 2 .) 



28. When the ring is very thin and narrow, the integrations can be performed at 

 once with reference to q>, and we find, by using a for the mean radius of the ring, 

 (280) 



