164, 165] STEADY FLOW IN CONDUCTORS. 331 



the same properties with the possible exception of certain sur- 

 faces 5 at which they may be discontinuous, then if the values 

 of V are given on parts of the surface S bounding the region T, 

 and the value of dV/dn is zero on the remainder, the integral J 

 throughout the region T, 



is a minimum* for that function V which makes the vector q 

 solenoidal, where 



3F ^dV 



(2) u=qcos(qa}) = \, v = q cos (qy) = X -- , 



w = q cos 



. 



For if we change the form of the function V by the arbitrary 

 amount SF, 



3(F+SF) 



T 



(3) = 



/T 



JJJ 



, 



dr. 



to 



\ dy J \ dz 

 The integral with the coefficient 2 is equal, by Green's theorem, 



ff[ 



"III 



JJJ 



dy \ dy J dz\ dz 



where n^ and n 2 are the normals on opposite sides of a surface S 

 of discontinuity of the derivatives. On those portions of the 

 bounding surfaces for which F is given 8F=0, and for the re- 



* Kirchhoff, Ges. Abh. p. 44. 



