260 Mr. J. Parker on the Theory of Magnetism and 

 throughout its volume. The density of the volume distri- 

 bution is therefore — -.-, where the differential coefficient is 



found on a line of magnetization. 



Now let a finite body be divided into an infinite number of 

 thin tubes such as that surrounding the curve OPQ. Let one 

 of these tubes meet the surface of the body in the curves 

 XY, X'Y', and draw two normal sections XZ, X'Z' to the 

 tube, entirely within the tube and just touching the curves 

 XY ; X'Y' at the points X, X'. Then, when the section of the 

 tube is indefinitely diminished, the external magnetic action 



Fig. 8. 



of XYY'X' is ultimately the same as that of XZZ'X', and is 



therefore equivalent to a volume-density whose value p at any 



point is —~r, and a layer of surface-density I x on XZ and 



another layer — I x/ on X'Z'. Consider the section XZ. The 

 layer on this section is equivalent to an equal quantity of 

 magnetism distributed uniformly on the neighbouring small 

 area XY. But if 6 X be the angle between the direction of 

 magnetization and the outward drawn normal at X, the area 

 XY=the area XZ x sec X . The surface-density on XY 

 is therefore I x cos 6 . Similarly the density on X'Y' is 

 I x / cos X ,. Hence we arrive at the simple result, generally 

 obscured or made mysterious by formidable integrations, that 

 a finite body is magnetically equivalent to a volume distribu- 

 tion whose density p at any point is — — , together with a 



surface-layer whose value a at any point of the surface is 

 I cos 0, where 6 is the angle between the direction of magneti- 

 zation and the outward drawn normal at the point. 



The expression for p can be put in a more convenient form. 

 For if three equal bodies whose magnetizations are respec- 



