84 Mr. J. Aitken on some Experiments on 



chain moves in a circular path, the centrifugal force and the 

 tension are evidently equal and balanced at all points ; and we 

 also know from experiment that, when the chain hangs in the 

 form of a long loop from a pulley, the tension just balances 

 the centrifugal force at all points, as the chain has no tendency 

 when in motion to alter the form of the loop. I shall not 

 attempt to enter into a mathematical investigation of this 

 balance between these forces ; my province is simply to describe 

 some experimental illustrations ; I would, however, refer all 

 those who wish for a mathematical investigation of the subject 

 to Thomson and Tait's ' Elements of Natural Philosophy,' where 

 they will find it fully treated. There is, however, an extremely 

 simple geometrical demonstration, which I may venture to give 

 before proceeding further. 



Let us consider the equilibrium of an endless chain moving 

 in a loop of such a form as that represented in fig. 7, PL VII., 

 the links of which are moving in a path of varying curvature. 

 As the velocity of the links is the same at all points, it will 

 not be necessary for us to consider how different velocities of 

 the links will affect the tension in the chain ; and the investi- 

 gation confines itself to the consideration of the tension pro- 

 duced in the chain by the links when moving in paths of 

 different rates and amounts of curvature. 



The first point to be considered is, What is the effect of the 

 rate of curvature of the chain when the angle of deflection is the 

 same ? Suppose A B C and D E F, fig. 1, PI. III., to be two 

 chains moving with the same velocity and in directions parallel 

 to each other, and suppose the radius of the curved part of the 

 path in D E F to be only one half of what it is in A B C. Now, 

 as the deflection is the same in both cases, the integral forces 

 required to produce these deflections will evidently be equal; 

 or (to state it in another way) the force required to destroy 

 the momentum in the chain in the direction A B, and to gene- 

 rate momentum in a direction at right angles to A B, is the 

 same in both cases. The whole resistance offered by all the 

 links in the bend to this deflection, or what is called the 

 centrifugal force, is therefore equal in both cases, and the ten- 

 sions produced in the two chains will therefore be equal. So 

 long, then, as the deflection is the same the different rates of 

 curvature produce the same tension in the chain, and have no 

 tendency to alter the path in which the chain moves. The only 

 difference between the two cases is, that the centrifugal force 

 in the curved part of the chain D E F is twice as great per 

 unit of length as in the curved part of ABC. Suppose, for 

 instance, that there are 10 links in the bend in D E F, and 

 that each link exerts a force of 20 units. Then, as the radius 



