38 GENERAL THEOREMS. 



are null at the point P , this point is a singular point of the level 

 surface V ; the force there is null, it is a point of equilibrium. 



For adjacent points we may neglect the powers of the 

 co-ordinates higher than the second, and the expression of the 

 potential V reduces to 



The equation 



represents a cone of the second degree tangential to the equi- 

 potential surface at the point of equilibrium P . 



If the function H 2 is itself identical with null, as well as some 

 of the following ones, and if H n is the first function of the 

 development, which does not vanish, the equation 



H =0 



will represent, in like manner, a cone of the n th degree, tangential 

 to the equipotential surface at the point P . 



This cone will be formed of n sheets, or of a smaller number, 

 corresponding to an equal number of sheets of the level surfaces. 



If the sheets of the cone do not intersect, neither do the 

 equipotential surfaces, and P is an isolated point of equilibrium. 



If the sheets of the cone do intersect, every line of intersection 

 is tangential to the intersection of the corresponding sheets of the 

 equipotential surface that is to say, a line of equilibrium which 

 passes through the point P . 



48. If the equipotential surface at the point P consists of two 

 sheets which intersect^ the intersection of the sheets takes place at a 

 right angle. 



When the equipotential surface consists of two sheets which 

 intersect, the cone of the second degree, which is tangential at a 

 point P of the line of intersection, reduces to two planes. If 

 we take the tangent to this line for the z axis, the equation of the 

 cone H 2 = will contain no term in z. 



In order to satisfy Laplace's equation, which reduces to 



<> 2 H 



o, 



the coefficients of the terms in x 2 and in y 2 must be equal and of 



