68, 69] Equipotentials and Lines of Force 59 



JEquipotentials which intersect themselves. 



69. We have seen that, in general, the equipotential through any point 

 of equilibrium must intersect itself at the po-int of equilibrium. 



Let x, y, z be a point of equilibrium, and let the potential at this point be 

 denoted by F . Let the potential at an adjacent point # + , y + y, z-\- f, be 

 denoted by F$, ,,, f . By Taylor's Theorem, if f(x t y t z) is any function of 

 x, y, z, we have 



-) 



where the differential coefficients of f are evaluated at x, y, z. Taking 

 f(x, y, z) to be the potential at x, y, z, this of course being a function of the 

 variables a?, y, z, the foregoing equation becomes 



If x, y, z is a point of equilibrium, 



ar = 8F = a 



da dy dz 



so that Ffc%f _F. + 



Referred to a, y, z as origin, the coordinates of the point 

 z + become f, 77, f, and the equation of the equipotential F= (7 becomes 



In the neighbourhood of the point of equilibrium, the values of f, 77, f are 

 small, so that in general the terms containing powers of f, 77, f higher than 

 squares may be neglected, and the equation of the equipotential V=C 

 becomes 



f a ^+ 2^|^- + ... = 2 (0- F ). 

 ' 9# 2 8^y 



In particular the equipotential F= F becomes identical, in the neighbour- 

 hood of the point of equilibrium, with the cone 



Let this cone, referred to its principal axes, become 



af /2 + &7/ 2 + < 2 = ........................... (26), 



then, since the sum of the coefficients of the squares of the variables is an 

 invariant, 



A 



= 1 + ^ + ^ =0. 



dx dy d2 



