INDUCED MAGNETISATION 239 



that if N is the component of the intensity normal to the 

 surface, 



/"NdS = 4-7T X quantity within. 



For the surface chosen N has value Hj over PR and H 2 over 

 QS, and is zero on the sides. If 6^ # 2 are the angles with the 

 normal made by H x H 2 , the cross-sections at PR and QS are a cos 9^ 

 and a cos 2 , so that we get 



Hj a cos O l H 2 a cos 9 2 = 47r<ra 



and o- = H i CQS 0i - H 2 cos 2 



4-7T 



We may express a- in terms of either H x or H 2 , since Hj cos 

 = B x cos 9 l = B 2 cos 9 2 ^H 2 cos $ 2 , whence 



= H 2 cos 2 . . 



It is easily seen that if p is constant within the surface the total 

 magnetism on one end of a tube where it enters the surface is 

 equal and opposite to that on the other end where it leaves. For 



(7 . PQ = <rQS/cos 9 2 = A - H 2 . QS, which is constant along the 



4f7T 



tube within the body. 



Magnetic susceptibility. In another way of viewing the 

 magnetisation of iron in air, the distribution 

 of polarity imagined as above is regarded 

 as the surface manifestation of the internal 

 magnetisation of the iron, and we may then 

 consider the surface layer cr as describing a M y 

 real physical condition of the iron beneath it. 

 Ixit PQRS, Fig. 185, represent a tube of in- 

 duction entering the iron at PQ, and leaving 

 it at RS. We shall suppose that JUL is constant 

 within the iron, so that magnetism need be 

 imagined only at the ends PQ and RS. We 

 may represent this magnetism by supposing 

 that the tube is divided into magnetised 

 fibres, each with poles only at PQ and RS. If we could cut through 

 these fibres at any intermediate section MN, perpendicular to 

 the axis of the tube, opposite polarities would be developed at the 

 two cut faces, each equal in total quantity to the polarity at either 

 end. If the surface density of magnetism on MN is I, MN X I is 

 constant along the tube and is equal to PQ X a-. If MN is close 



