360 Permanent Magnetism [OH. xi 



If the position of the surface S is determined by geometrical conditions 

 if, for instance, it is the boundary of a small rectangular element dxdydz 

 then we cannot suppose it to contain only complete magnetic particles, and 

 equation (334) will not in general be true. 



If there is no magnetic matter present in a certain region, equation (334) 

 is true for any surface in this region, and on applying it to the surface of the 

 small rectangular element dxdydz, we obtain, as in 50, 



(33o), 



the differential equation satisfied by the magnetic potential at every point of 

 a region in which there is no magnetic matter present. 



Tubes of Force. 



409. A tubular surface bounded by lines of force is, as in electrostatics, 

 called a tube of force. Let o^, o) 2 be the areas of any two normal cross-sections 

 of a thin tube of force, and let H lt H z be the values of the intensities at 

 these points. By applying Gauss' Theorem to the closed surface formed by 

 these two cross-sections and the portion of the tube which lies between 

 them, we obtain, as in 56, 



provided there is no magnetic matter inside this closed surface. 



Thus in free space the product Hco remains constant. The value of this 

 product is called the strength of the tube. 



In electrostatics, it was found convenient to define a unit tube to be one which ended 

 on a unit charge, so that the product of intensity and cross-section was not equal to unity 

 but to 47T. 



Potential of a Magnetic Particle. 



410. Let a magnetic particle consist of a pole of strength m^ at 0, and 

 a pole of strength +m l at P, the distance OP being 



infinitesimal. 



The potential at any point Q will be 



n Q =-^-^ (336). 



If we put OQ = r, and denote the angle POQ by 6, - mi ' mi 



this becomes Fia. 105. 



n m^OQ-OP) mi OP cos fjLCOsd /Qfm 



^~ PQ.OQ PQ.OQ ~^~ 



where //, = r^ . OP, the moment of the particle. 



