§ 29] THE SATURNIAN SYSTEM. 19 



2<Q 



'J* J 



_m_ n - 2M f 1 



2'0J0/ " < 5W[(a + r)» + * 2 ] " 



2© 







= * (F+ r2 - a2 -° 2 EA 





2© 



E 1 



These formulae give 



-©[(a-rj' + ^V [(« + *•)* + **] 



29. If, in the general case, a plane is drawn through the attracted point perpen- 

 dicular to the axis of the cylinder, and if, in this plane, two circles are taken, with their 

 common centre in the axis, and with radii / and r", connected by the equation, 



in which q is the distance of the common centre from the attracting particle (o, £), 

 every point in either circumference has the same value of i, which may be determined 

 by the equation 



. r' + r" — 2 e 



cos'*= ' „ \ 



r-\-r -\-2q 



The two circles may be called complementary, and are derived from each other by a sim- 

 ple and obvious geometrical construction. It is apparent, indeed, that they are both 

 tangent to a spherical surface, which passes through the particle (o, t), and is tangent 

 to the cone having for its axis the axis of the cylinder, for its vertex the common cen- 

 tre of the circles, and which passes through the point (o, c). 



If the values of § 29 are, then, denoted as functions of / and r", they give 



n (/') = y£ a (/) = l„ n (/) = ± a (/), 



R (O = 4* -Q ('0 - 4 » (O = 4 [ R (0 - ^ z ('•')]• 



r r r L * J 



(281) 



