INTENSITY OP THE EARTH'S MAGNETIC FORCE. 255 



The ratio of these quantities is independent of m', and we have 

 simply 



*'-,,. 



TT 



Finally, substituting in the value of h above given, and designat- 

 ing the half lengths of the deflecting and the suspended magnets 

 "by / and /', respectively, 



h = f (2l 2 - 3r 2 ) ; 



a quantity whose value may be exactly known, independently of 

 experiment. This quantity vanishes, when P = f I' 2 , or 

 J- 1-224 /'; 



a result which is independent of the magnetic state of the bars.* 



As the preceding results depend, in part, upon an empirical law 

 of magnetic distribution, which is only approximately true in the 

 case of small magnets, it seemed desirable to obtain a confirmation 

 of their accuracy by direct experiment. The nature of such con- 

 firmation will be immediately understood from the form of the 

 relation between the angle of deflection and the distance. For 

 since, in the method of deflection employed by Gauss, DHanw 

 = Q (1 + /2D~ 2 ), the function D 3 tan u will be constant for all values 

 of D, when h = ; while, if the coefficient of the fifth power of the 

 distance has a sensible value, it will vary with D, its values form- 

 ing a decreasing or increasing series, as D increases, according as h 

 is positive or negative. Hence we have only to observe the deflec- 

 tions produced at different distances, when the two magnets have 

 the relative lengths pointed out above, and to compare the results 

 with those obtained under other circumstances. 



Several series of experiments were accordingly made in the be- 

 ginning of the present month, in some of which the lengths of the 

 two magnets were the same, while in others they were in the 



* If the centre of the deflecting magnet be in the magnetic meridian passing through 

 the centre of the suspended magnet, and its axis perpendicular to th same line, we find, 

 by a process similar to that above given, that the condition to be fulfilled, in oider that 

 the term involving the fifth power of the distance may vanish, is 



*!_ 4 *L. 



M M' ' 



and, accordingly, that the corresponding relation between the lengths of the two magnets 



is, in that case. 



/= 2f. 



