152, 153] CONDITION FOR EQUIPOTENTIAL FAMILY. 411 



If this is satisfied, we have 



dV i 



~dq. = ~J 



5) V = A J' 



There must be one value q such that the level surface is a sphere 

 of infinite radius, and for this F must vanish. 



These conditions are satisfied by the polar coordinate r, for by 

 141, 87) 



2 



Ar 



V = A I 



= A^ 



For r = oo, we must have V = 0, accordingly we must put B =Q. 



We may get a convenient expression for | by transforming z/# 



h q 



into terms of three orthogonal coordinates, of which it is itself one. 

 Put q = g 1; and since it is independent of q 2 and # 3 , 



6) 



7^> ^gt = h * h * d \ ^ V 



" 



153. Application to Elliptic Coordinates. Applying this 

 to elliptic coordinates gives 



