52 MEASUREMENT OF COUPLES. 



In his experiments on Terrestrial Magnetism, Gauss* fixed to the 

 movable arrangement a horizontal divided scale, and as auxiliary 

 body used two spheres which he suspended to the scale by hooks, at 

 equal distances on either side of the axis. The distance d of the 

 points of attachment of the two hooks is accurately measured. If q' 

 is the weight of each hook, p its radius of gyration in respect of an 

 axis passing through the centre of gravity of the sphere, a the radius 

 of the spheres, and q the weight of each of them, the value of the 

 moment of inertia of the additional system is 



the term q^ may usually be neglected. 



If we desire to eliminate the moment of inertia Kj of the scale, 

 the system is made to oscillate in three different conditions ; ist, 

 without the scale ; 2nd, with the scale and without the spheres ; 3rd, 

 with the scale and the spheres. The equations 



enable us to determine the three quantities, K, K 15 and C, as a 

 function of the moment of inertia K', and of the three times of 

 oscillation, T, T lt and T 2 . It is moreover useful to repeat the third 

 determination with different values of d, and to calculate the un- 

 known quantities by suitable combinations of all the equations. 



A rectangular plate, a sphere suspended to an axis, a solid or 

 hollow cylinder, would perform the same office with more or less of 

 advantage. 



704. There is, in practice, always reason to fear that, whatever be 

 the mode of suspension of the movable system, the directive couple 

 is not independent of the weight. 



The experiments are then made in such a way as to modify the 

 moment of inertia of the system without altering the total weight. 

 The oscillating arrangement must be composed of a fixed part, whose 

 moment of inertia is K, and of a movable part, the moment of 

 inertia of which acquires known values K' and K", in two successive 



* GAUSS. Intensitas vis Magnet, etc. Comm. Gottin., vm. 1841. (Euvres, 

 v., p. 95- 



