NEW DlAMAGNETIC ATTACHMENT TO THE LANTERN 77 



and the other components of the force are 



and 



- 



Let, now, the axis of z be vertical, the axis of x in the line of the 

 magnetic poles of the magnet, and y at right angles to both. Then 

 the moment of the forces acting on the body to turn it about the axis 



of z is 



where the integration extends throughout the volume of the body. 



If the body is suspended so as to turn freely about the axis of z it 

 will vibrate about the position for which M is a minimum or else will 

 remain at rest at that point. The number of single oscillations made 

 when the angular elongation & is very small, is 



1 / M 



' T. V tfj' 



in which M and $ must be measured simultaneously, and I is the 

 moment of inertia of the body. 



I r r r 



A/ I l/f 

 \ J J J 



i Jw d(i^)\, ^ ^ 



y , 3 -, \dxdydz. 

 \ J dx dy j 



Xow let us suppose that the whole apparatus changes size, the relation 

 between the parts remaining constant, so that the apparatus becomes 

 m times as great as before. Then x, y, dx, dy, and dz will increase ra 



times and /, m 5 times. To determine the changes in ^ ^ and -X * 



aye? ^y 



we make use of the theorem of Sir Win. Thomson, that " similar bars 

 of different dimensions, similarly rolled, with lengths of wire propor- 

 tional to the squares of their linear dimensions, and carrying equal 

 currents, cause equal forces at points similarly situated with reference 

 to them." But as the above only applies to equal currents, I have 

 generalized it in the following: In any two magnetic systems whatever, 

 similar in all their parts and composed of any number of permanent or 

 electro-magnets, wires carrying currents, or bodies under magnetic induc- 

 tion, the magnetic force at similar points of each will be the same when the 

 following conditions are complied with: 1st, the magnetic materials at 

 similar prints in the two systems must be exactly the same in quality and 



