128 PARTICULAR CASES OF EQUILIBRIUM. 



consists of two symmetrical branches, starting from A and A' tan- 

 gentially to the vertical, and which are asymptotes to a straight 

 line OL. 



In order to determine the direction of the asymptote, let us 

 consider a very distant point ; then, if 8 is the very small difference 

 <o - co', and observing that the angles co and co' ultimately become 

 equal, 



cos co cos to' cos co' cos co sin co. 8 sin co r8 



' 2 r't rt 2r(r' r) 2r 2 r' r 



sin to. 2a sin to i sin 2 to 



whence 



tan 2 co=2. 



148, PRINCIPLE OF IMAGES. We have already seen (59) that 

 we can always replace any mass of electricity by an equal mass 

 distributed over an equipotential surface which completely sur- 

 rounds it. This layer is of itself in equilibrium, and its density is 

 defined by the condition 



For all points in the interior, the potential becomes constant and 

 equal to that of the surface ; but for all external points, nothing is 

 changed in the state of the field. 



Let us consider the plane Oy at potential zero (Fig. 34) in the 

 preceding problem. For all points on the right we may replace the 

 mass m on A' by an equal mass in equilibrium on the plane. The 

 density will be P 2 at each point, the force F 2 being directed towards 



the left, 



F 2 (21110) i 



4?r 477 /o 3 



We see thus that it is inversely as the cube of the distance of the 

 point in question P 2 from the point A. 



It may be observed, from this law of distribution, that the charge 

 of an element of the plane is everywhere proportional to the angle 

 which it subtends at the point A. In fact, the charge of a surface 

 element dS (Fig. 36) is 



27T 



