THE INVERSE SQUARE LAW 209 



vibration be t when the earth alone acts, and ^ and t 2 when the 

 pole is at the distances d t and d 2 from the needle. Regarding the 

 needle as a sort of double pendulum with equal and opposite forces 

 acting at the two ends, it is seen that the times of vibration are 

 inversely as the square roots of the forces acting : 



H : H + F x : H + F 2 =^ 2 : 1 : i. 



*1 r 2 



whence the forces due to the magnet alone are given by 



F - F 1 1 - - 1 - l 

 2 t* t* ' t* I 2 ' 



By observing the times of vibration Coulomb found that the 

 forces were approximately proportional to the inverse square of the 

 distances c^ and 6? 2 , or that the force due to a given pole is inversely 

 as the square of the distance from that pole. 



He also verified the law by means of the torsion balance (see 

 ante, p. 62), using two long wire magnets about the same length 

 as the one described above, and about J inch in diameter, one being 

 suspended horizontally in the torsion balance, the other being fixed 

 vertically so that if the suspended one had remained in the 

 magnetic meridian, their two like poles, taken as being one inch 

 from the end of each, would have been in contact. The suspended 

 needle was repelled from the magnetic meridian, and the torsions 

 were then measured which were required to bring the two poles to 

 observed distances from each other. In each case the action of 

 the earth aided the torsion, and a preliminary experiment was 

 necessary to determine the torsion equivalent to the earth's 

 action. Thus in one experiment, before the vertical magnet was 

 put in position, two turns of the torsion head, i.e. 720, pulled the 

 magnet round 20, or 700 torsion were required to balance the 

 earth's couple due to 20 deflection from the meridian, giving 35 

 of torsion per 1 of deflection. Putting the vertical magnet in 

 position and bringing the torsion head back to its original position, 

 the suspended magnet was deflected 24, the torsion being therefore 

 24. To this we must add 24 x 35 = 840 for the torsion 

 equivalent to the effect of the earth, making a total of 864 at 

 24. To halve the deflection, the torsion head had to be turned 

 eight times round, giving a torsion of 8 X 360 + 12 = 2892. To 

 this must be added 12 X 35 = 420 for the torsion equivalent of 

 the earth's effect, making a total of 3312 at 12. According to 

 the inverse square law, with this value at 12, taking distances as 



3312 

 equal to arcs, we ought to have at 24 a torsion of ~ = 828, 



which is not very far from the observed value 864. 



Magnetic measurements founded on the law. We 

 shall describe below a still more accurate method of verifying 



