If, then, 

 let 



2 THE SATURNIAN SYSTEM. [§ 1 



z = the distance of the attracted point from the plane of the ring, 

 L = the distance of the particle from the plane of the ring, 

 / = the distance of the attracted point from the particle. 



The following notation is also introduced into this memoir : * 



j = v/-i, 



*© = the ratio of the circumference to the diameter of the circle, 

 = 3.1415926536, 



& = the base of Naperian logarithms, 



= 2.7182818285, 

 which gives 



v /@' 3 = J- J = 4.810177381. 



F = any function of q and c = F (q, L) 



F 2 =F(o, b), F =P( 9 , 0), F, =F(o, -h), 



F 2m = F (a m , b), Fc = F (a m , 0), F lm = F («„„ -b), 



F 20 = F (0, b), Foo = F (0, 0), F 10 = F (0, -b), 



F ;o =F(0, £), 



F (0) — the value of F when s vanishes, 



F (0) = the value of F when r vanishes, 



F[oj = the value of F when r and s vanish. 

 Let, moreover, 



k = the density of the cylinder, 

 do = q dq d(p d £ = the volume of a particle, 

 dy = n d r> d(p, 



£2 = k I — = the potential of the ring for the attracted point 

 The value of/ is given by the equation, 



f = r 2 + ^ 2 — 2 r q cos tp -j- (a — t ) 2 



= (>" + 'v') 2 s ' n ' J 2?"f (>' (,0 2 cos2 a f jP + ( g £) 2 



= ('• + 9 f — 4 '' Q co * 2 * 9 + (* — C) 2 



= (9 — t cos f r ) 2 -\- > 2 sui ' 2 ( / j -f~ ( g — O 2 - 



* This nutation is introduced in order to set free the letters i, ir, and e, which are so much wanted for astronomical pur- 

 poses. The sign for ^ — 1 is similar in aspect to i, and is nearly an inverted r. The sign for the ratio of the circumference to the 

 diameter is nothing more than a c with the lower extremity extended and wound over the letter; while that for the Naperian base 

 is nearly the letter 6 with the initial extremity bent down over the symbol. 

 (264) 



