254 ON THE DETERMINATION OF THE HORIZONTAL 



In this case, therefore, ^ = 90, and the equation of equilibrium is 



reducedt ajf (l /.* ,'\ij 



Xsme, -l + -a-, 



and the equation is the same as that to which it is reduced in the 

 former case, the sine of the angle of deflection being substituted 

 for the tangent. It appears from the result, that this method is to 

 be preferred to the former, not only because the angle of deflection 

 is greater, ccet. par., but also because the variable part in the 

 coefficient of the inverse fifth power of the distance is strictly 

 evanescent. 



It remains now to inquire in what manner the quantity h, which 

 expresses the ratio of the two coefficients, may be known d priori ; 

 and whether that quantity can be made to vanish, by any simple 

 relation between the acting magnets. 



For this purpose we must know, at least approximately, the law 

 of magnetic distribution, or the function of r by which m is repre- 

 sented. Almost the only knowledge which we possess on this sub- 

 ject is that derived from the researches of Coulomb. From these 

 researches M. Biot has inferred that the quantity of free magnet- 

 ism, in each point of a bar magnetized by the method of double 

 touch, may be represented by the formula 



/u being a quantity independent of the length of the magnet, and 

 A a function of ju and /. M. Biot has further shown, that when 

 the length of the magnet is small, the relation between m and r is 

 approximately expressed by the simple formula 



= 



the curve of intensities becoming, in that case, very nearly a right 

 line passing through the centre of the magnet. 



Employing, then, this approximate formula, we have 



r+l r r-+i 



M = J ^ mrdr = | r z dr = f w'/ 2 ; 



r+f m ' r+i 



M 3 = I mt*dr - j- r'dr = f m' I*. 



