116 



The following modification of one of the methods pro- 

 posed by Gauss, for the attainment of this end, appears to 

 combine the greatest number of advantages ; namely, to take 

 three readings, at the times 



t — T, t, < + T ; 



t being the epoch for which the position of the magnet is de- 

 sired, and T its time of vibration.* In order to show that 

 tbis method is adequate, it is necessary to deduce the' 

 equation of motion of a vibrating magnet ih a retarding 

 medium. 



Let X denote the horizontal part of the earth's magnetic 

 force ; q the quantity of free magnetism in the unit of vo- 

 lume of the suspended magnet, at the distance r from the 

 centre of rotation ; and 6 the deviation of the magnet from 

 its mean position. The moment of the force exerted by the 

 earth on the element of the mass, dm, is 

 -Kqrdm sin Q ; 



and the sum of the moments of the forces exerted upon the 

 entire magnet is 



Xju sin Q ; 



where fx denotes the value of the integral § qrdm, taken 

 between the limits r = ± ?, 21 being the length of the 

 magnet. 



Again, the velocity being small, the resistance may be 

 assumed to be proportional to the velocity. Accordingly, if 

 w denote the angular velocity, the retarding force due to 

 resistance, upon any element of the surface, ds, at the dis- 

 tance r from the centre of motion, is 



— K Js rw ; 



and the entire moment of this force upon the whole mag- 

 net is 



* In practice, it is sufficient to take tlie nearest whole number of seconds, 

 for the value of t. 



