INTENSITY OF THE EARTH' S MAGNETIC FORCE. 253 



M denoting the deviation of the axis of the magnet from the 

 direction of the force. Multiplying by clr, therefore, and in- 

 tegrating, the total moment is 



XM'smu. 



Hence the equation of equilibrium is 



There are two cases of this solution which demand our con- 

 sideration. 



In the method of Grauss, the deflecting magnet is perpendicular 

 to the magnetic meridian, and therefore i// = 90 - u. In this case, 

 then, the preceding equation becomes 



Accordingly, the term containing the fifth power of the distance is 

 composed of two parts, one of which is constant, while the other 

 varies with the angle of deflection; so that, if there were no 



11 f- li /"' 



means of determining d priori the values of the ratios -ri -=^-, 



JU. Ju. 



three equations of condition would be, in strictness, required for 

 the determination of the three unknown quantities ; namely, the 

 coefficient of the inverse cube of the distance, and the two parts of 

 the coefficient of the inverse fifth power. However, the distance 

 being greater than four times the length of the magnet, the angle 

 of deflection, it, is always small, and the term involving the square 

 of its sine may be neglected in comparison with the others. 

 Accordingly, if we make, for abridgment, 



2Jf M 3 Jf,' 



the expression for the tangent of the angle of deflection is reduced 

 to the form 



In the method of deflection employed by Professor Lamont, 

 the deflecting bar is always perpendicular to the suspended bar. 



