144 



PARTICULAR CASES OF EQUILIBRIUM. 



In fact, for any point at a distance r in the direction w, the angle 

 6 of the resulting force with the radius vector r^ is given by the 

 ratio 



tan (9 



sin w - ~F t 

 cos w + F_ 



tan to 



2- 



This angle is always null when r = a that is, when the point is 

 on the sphere. For all points on the equator, however, the angle w 



is equal to -, and the expression assumes an indeterminate form. 



2 TJ. 



Let us suppose that the angle o> is very little different from -, the 

 angle 6 for a point P (Fig. 42) near the surface is 



r a 



a tan ( o> 



2 



r- a 



r a 



i 

 tan 



Fig. 42. 



Multiplying this equation by the preceding, which always holds, 

 and observing that the difference r - a is very small, we get 



a r^ # 3 a 30* (r a) 30? 

 tan 2 6 = . - = . = = :=i. 



r-a 



a 

 r-a 



+ 20? r^ + 20? 



The lines of force which touch the sphere on the equator make, 

 therefore, with the surface, an angle of 45. 



The equipotential surface at the original potential V of the 

 centre of the sphere is a plane which terminates at the equator, and 

 is thus prolonged by the surface of the sphere itself. The equator is 

 a line of equilibrium. 



