406 Prof. W. Cassie on the 



in practice, however, the centre of gravity does not occupy 

 exactly this position, the effect of a small error in this respect 

 proves to be so small that it may usually be neglected. In 

 any case it can be allowed for. The calculations are given 

 below. The position of the centre of gravity of the central 

 tube can be separately adjusted to the right level by means 

 of the gravity bob, Gr. fig. 4, which is attached to the tube 

 for that purpose. And as the weight of the central tube is 

 a very small fraction of the whole weight of the needle, 

 any outstanding error in the position of its centre of gravity 

 would be quite inappreciable in its effect. 



In the bifilar oscillation the period is affected by the 

 resistance to torsion of the wires. This effect is eliminated 

 by observing the periods with the needle suspended by first 

 the inner, and second the outer pair of knife-edges, so that 

 the suspending wires are at two different distances apart. 

 The supporting pulleys on which the wires are clamped at 

 the top are made of diameters equal to the distances between 

 the knife-edges, so that the wires are parallel in each 

 experiment. The free lengths of the wires are taken the 

 same in each experiment ; this can be secured automatically 

 by an appropriate arrangement of the pulleys. 



Calculation of the Periods. — Of the oscillations possible to 

 this system we shall make use of three. They are rotations 

 about three perpendicular axes through the centre of the 

 needle, viz. : — 



1. About a vertical axis — the bifilar oscillation. 



2. About a horizontal axis perpendicular to the length of 

 the needle — pitching. 



3. About a horizontal axis along the length of the needle — 

 rolling. 



Let I be the length of each of the wires, 

 M the mass of the needle, 



X the modulus of stretching of each of the wires, 

 t the modulus of torsion of each of the wires, 

 k 2 the radius of gyration of the needle about its axis of 



figure, 

 h the radius of gyration of the needle about an axis 

 perpendicular to its axis of figure through the 

 centre of gravity. The needle is made a figure of 

 revolution so that this radius of gyration may be 

 taken the same in all such directions. 



Firstly, assume that the mass of the hooks on which the 

 knife-edges rest may be neglected in comparison with that of 

 the needle, and that the centre of gravity of the needle is at the 



