2 Stephenson, New Type of Dynmnical Stability. 



integer.* The intensity of the magnifying effect falls off 

 rapidly when r is taken larger, and as our terrestrial 

 systems are always subject to friction, it is difficult to 

 exhibit many cases experimentally. With the pendulum 

 the intensifying influence may be observed for several 

 values of r without special nicety of adjustment. 



Our present object is to establish another very remark- 

 able property of the non-generating type of periodic 

 disturbance. 



2. In the preceding pendulum experiments the result- 

 ing motion is due to the combined action of the imposed 

 force and gravity. Let us enquire as to the effect of the 

 imposed force acting alone. 



To examine this question experimentally, a rod is 

 pivoted vertically so that it is free to rotate in a horizontal 

 plane : when the pivot is driven horizontally in a simple 

 vibration along the length of the rod the relative equili- 

 brium is not disturbed. If, now, the rod is displaced 

 through a small angle, it is observed to swing about the 

 line of the imposed motion in a period large compared 

 with that of the pivot. 



All the properties of the motion may be deduced from 



the differential equation determining it, but here we seek 



an approximate treatment of the problem, based on 



general mechanical considerations, in order to obtain a 



notion of what happens more vividly than is possible 



from the exact solution. For this purpose we assume 



that a, the amplitude of the pivot motion, is small, and 



also that the speed of the pivot, P, is constant, and equal 



to F, say, throughout the path ; i.e., we assume that the 



body is acted on by suitable impulses applied at P at the 



ends of its path, being free from action in all intermediate 



* " On a class of forced oscillations," Quart. Journ. Math , No. 148, 

 1906, 



