§ 2-5] THE SATURNIAN SYSTEM. 3 



2. The integration with reference to t gives, by familiar forms of integration, 



v>^n = k{n' 2 -ir), 



in which 



I2' = log(/_ g + Q. 



3. When the ring is very thin, and the attracted point nearly in its plane, so that 

 the second dimensions of z and b can be neglected, the formulas give 



f = /o(0) 



r— * 



J2' = log/. 

 £2' 2 = \ogfJ- 



f ' 



b—z 



~7~' 



(v — i™ -f H 



K = log/ 



so that the result is the same as if the values were reduced to 



4. In the same way, when the attracted point is so far from the cylinder, that the 

 cube of b can be rejected in comparison with the square of the distance of the point, 

 the formulas give, by reduction, 



/•=/-- ? +-Mi--) 



J /o /. ^ 2./; V f*J> 



q>_cv — 2 J 



Jo 



which is the same as if the values were reduced to 



£1' — £~l' ■=.—. 



Jo 



5. The integration with reference to q gives, by introducing 



£2" =f p (,>£?), 



94 =f p (o £2' 2 ) =f p [ 9 log (/ 2 -» + *)] 



= I </ log (/ 2 - + b) - i/ p [o 2 D p log {U—z + 5)] 



(265) 



