374 Permanent Magnetism [CH. xi 



distribution on the ends is one of density + / cos 6. Thus the force arising 

 from distribution III is a force 47r/cos0 in the direction of the axis of 

 the cavity. 



Thus the force on a pole placed inside this cavity may be regarded as 

 compounded of the force H (arising from distributions I and II), and a force 

 47r/ cos 6 in the direction of magnetisation, arising from distribution III. 



Let e be the angle between the direction of the force H and the axis of 

 the cavity, then the component force in the direction of the axis of the cavity 



= H cos e + 4-TrJ cos 6. 

 If I, m, n are the direction -cosines of this last direction, 



H cos e = lot + mf} + ny, 

 47r/ cos 6 = 4?r (I A + mB + nC), 

 so that, by equations (359), 



H cos e + 4-7T/ cos 6 = la + mb + nc. 



Thus the component of the force in the direction of the axis of the cavity 

 is the same as the component, in the same direction, of the magnetic in- 

 duction, namely la + mb + nc. 



434. We are now in a position to understand the importance of the 

 vector which has been called the induction. This arises entirely from the 

 property of the induction which is expressed in the following theorem : 



THEOREM. The surface-integral of the normal component of induction, 

 taken over any surface whatever, vanishes, 

 or in other words (cf. 177), 



The induction is a solenoidal vector throughout the whole of the magnetic 

 field. 



To prove this let us take any closed surface S in the field, this surface 



FIG. 110. 



cutting any number of magnetised bodies. Along these parts of the surface 

 which are inside magnetic bodies, let us remove a layer of matter, so that the 

 surface no longer actually passes through any magnetic matter. 



