CYLINDRICAL MAGNETS. 393 



triangle CAA' (Fig. 87), the base of which is equal to twenty-five 

 times the diameter. The angle a of the right line which represents 

 the densities is constant for bars which only differ in length. The 

 quantity of magnetism is then constant, and is the same as in a 

 limited magnet, for which we should have L = 50^; this quantity 

 may then be represented by a (50^) 2 and the moment by 



Coulomb, however, only considered these results as a first 

 approximation. He observed that if we take equidistant parts from 

 the end A of a magnet, the successive tangents to corresponding 

 points of the curve make with each other equal angles. The curve 

 which satisfies this condition is given by the equation e~^ = cos fix, 

 which for small values of x merges into an arc of a parabola CB 

 (Fig. 87) tangential to the axis at a point C, at a distance /, from the 

 end ; the quantity of magnetism is then proportional to / 3 , that is / 3 , 



and the pole is at a distance from the end equal to -. The magnetic 



/ A 4 



moment has the value ( L \ bl z . 



Fig. 87. 



It will be seen that the magnetic moment for a very long cylinder 

 tends to become proportional to the length, as in the case of induced 

 magnetisation. 



420. EMPIRICAL FORMULAE. These two portions of a parabola 

 do not represent the distribution of magnetism by a continuous 

 function. Biot found that Coulomb's experiments are represented 

 very exactly by the exponential formula 



(3) 



in which y is the magnetism at a point at a distance x from one end, 

 a and /* are constants. 



