60 Electrostatics Field of Force [CH. n 



Now a -f 6 + c = is the condition that the cone shall have three per- 

 pendicular generators. Hence we see that at the point at which an 

 equipotential cuts itself, we can always find three perpendicular tangents to 

 the equipotential. Moreover we can find these perpendicular tangents in an 

 infinite number of ways. 



In the particular case in which the cone is one of revolution (e.g., if the 

 whole field is symmetrical about an axis, as in figures 16 and 20), the 

 equation of the cone must become 



where the axis of f is the axis of symmetry. The section of the equipoten- 

 tial made by any plane through the axis, say that of f "C, ', must now become 



fs_2' a = 



in the neighbourhood of the point of equilibrium, and this shews that the 

 tangents to the equipotentials each make a constant angle tan" 1 \/2 (= 54 44') 

 with the axis of symmetry. 



In the more general cases in which there is not symmetry about an axis, 

 the two branches of the surface will in general intersect in a line, and the 

 cone reduces to two planes, the equation being 



ap + &7/ 2 = 0, 



where the axis of f is the line of intersection. We now have a + b = 0, so 

 that the tangent planes to the equipotential intersect at right angles. 



An analogous theorem can be proved when n sheets of an equipotential 

 intersect at a puint. The theorem states that the n sheets make equal 

 angles TT/H with one another. (Rankin's Theorem, see Maxwell's Electricity 

 and Magnetism, 115, or Thomson and Tait's Natural Philosophy, 780.) 



70. A conductor is always an equipotential, and can be constructed so as 

 to cut itself at any angle we please. It will be seen that the foregoing 

 theorems can fail either through the a, b and c of equation (24) all vanishing, 

 or through their all becoming infinite. In the former case the potential near 

 a point at which the conductor cuts itself, is of the form (cf. equation (25)), 



F 

 ,= 



dx 

 so that the components of intensity are of the forms 



8F 



-*(&+**-) 



The intensity near the point of equilibrium is therefore a small quantity of 

 the second order, and since by Coulomb's Law R = 47rcr, it follows that 



