STUDIES ON MAGNETIC DISTRIBUTION 103 



this whole quantity the part which affects the relative distribution at 

 any part of the rod most is that of the medium immediately surrounding 

 that part; and so the parts near the end have the advantage over those 

 further back, inasmuch as the lines can pass forward as well as outward 

 into the medium. The same thing takes place in the case of the dis- 

 tribution of electricity, where the "density" is inversely proportional 

 to the resistance which the lines of inductive force experience from 

 the medium; and here we find that the "density" is greatest on the 

 projections of the body, showing that the resistance to the lines of in- 

 duction is less in such situations, and by analogy showing that this 

 must also be the case for lines of magnetic force. But this effect is 

 not very great in cylinders until quite near the end; for Coulomb, in a 

 long electrified cylinder, has found the density at one diameter back 

 from the end only 1-25 times that at the centre; and so there is prob- 

 ably a long distance in the centre where the density is sensibly constant. 

 Hence we may suppose that our second hypothesis, that R' is a con- 

 stant, will be approximately correct for all parts of a bar except the 

 ends, though of course this will vary to some extent with the distribu- 

 tion of the lines in the medium; at least the change in E' will be 

 gradual except near the end, and so may be partially allowed for by 

 giving a mean value to r. 



Hence we see that could the formula be so changed as to include 

 both the variation of R and of R', it would probably agree with the 

 three Tables given. 



To study the effect of variation in the permeability more carefully, 

 we can proceed in another manner, and use the formulae only to get 

 the value of r at different parts of the rods. 



No matter how r may vary, equations (2) and (3) will apply to a very 

 small distance Z along the rod; and as the orgin of coordinates may be 

 at any point on the rod, if Q r and Q' f are taken at one point and Q and 

 Q t at another point whose distance from the first is Z, we shall have the 

 four equations 



Calling " =H and ? = G, we shall find, on eliminating C and A 

 and developing r ' and ?~ rt , 



