OF ARTS AND SCIENCES. 231 



The equipotential surfaces within the shell are quadrics, and it is 

 easily seen that they are hyperboloids. For if a, b, c denote their 

 semi-axes, we have 



„-, + 1+1 = 1 {<>».+ e.. + <>. } - 2 Qp-M {i + i + i} 



So that any internal equipotential surface is an hyperboloid of one 

 sheet of which the square of the imaginary axis is numerically the 

 least, or an hyperboloid of two sheets of which the square of the real 

 axis is numerically the least. The equations of the external equi- 

 potential surfaces involve elliptic functions. 



