441-444] Vector-Potential 381 



so that the most general solution of equations (371) is 



(375). 



, 

 dy dz 



Substituting these values, the line integral is found to be 



and the condition that this shall be equal to the surface integral is that 



J ~ds 

 or that x shall be single- valued. 



Thus if % is any singled- valued function, equations (375) represent a solu- 

 tion, and the most general solution, of equations (371). 



VECTOR- POTENTIAL. 



443. The discussion as to the transformation from surface to line inte- 

 grals arose in connection with the integral I INdS or \\(la + mb + nc) dS, in 

 which a, b, c are the components of magnetic induction. Since the condition 



is satisfied throughout all space, it must always be possible to transform the 

 surface integral into a line integral by a relation of the form 



f g+G ^ + H d )ds (376). 



The vector of which the components are F, G, His known as the magnetic 

 vector-potential. 



We shall calculate the values of the components of vector-potential in a 

 few simple cases. 



Magnetic Particle. 



444. Let us first suppose that the field is produced by a single magnetic 

 particle at the point x', y', z in free space, parallel to the axis of z. Then, 



by equation (338), H = p^-, f-J, so that at any point x, y, z, 



811 _ 8 2 /1\ 3 2 l 



~dx~~ ~ 

 and similarly 



