JAKUABT 20, 1911] 



SCIUNGE 



81 



representation, or the type of the oscilla- 

 tion may completely change; when the 

 latter happens, it is generally necessary to 

 change also the symbolic representation. 



A change in the character of an oscilla- 

 tion which does not cause real instability 

 can be illustrated by a simple example. 

 This illustration, while it has no immedi- 

 ate bearing on the question of synchron- 

 ism, does show the change in the character 

 of the motion which is sometimes produced 

 by a synchronism. 



Let us consider the motion of a rod, one 

 end of which is pierced to admit a hori- 

 zontal axle so that the rod can rotate in a 

 vertical plane. Suppose that the rod is 

 rotating so rapidly in one direction (called 

 positive) that it makes many complete 

 revolutions in a second. There will be 

 slight differences of velocity at various 

 points, differences which can be expressed 

 quite accurately by a single harmonic term. 

 Owing to frictional resistances the speed 

 will gradually diminish. With diminished 

 velocity the difference between the veloci- 

 ties at the highest and lowest points in- 

 creases in a ratio which varies very nearly 

 in the inverse ratio of the average speed. 

 When the speed has so far decreased that 

 the ratio of the velocity at the top to that 

 at the bottom is very small, we can no 

 longer express the differences of velocity 

 approximately by a single harmonic term, 

 but must include the higher harmonics. 

 At the critical stage when the velocity at 

 the top is just zero, the representation fails. 

 From our knowledge of the physical side 

 of the problem we know that this failure 

 is not owing to the defects of the repre- 

 sentation, but that it is due to a change in 

 the character of the motion. The rod 

 ceases to make complete revolutions and 

 begins to oscillate to and fro with diminish- 

 ing amplitude until it finally comes to rest 

 at the lowest point. There is another side 



to this example which is important for 

 what follows. The difference between the 

 velocities at the highest and lowest points 

 is continually increasing, but it does not 

 become infinite. At the critical stage it 

 reaches a maximum value and, becoming 

 discontinuous at this point, has its range 

 suddenly doubled. The minima and max- 

 ima are nearly equal in magnitude but of 

 opposite signs. The range then diminishes 

 until it reaches the limit zero. The critical 

 point is, of course, a position of unstable 

 equilibrium and then any small force which 

 may be acting, but which could previously 

 be neglected, may determine the character 

 of the future motion. 



If we neglect the resistance of the air 

 and imagine that the axle on which the rod 

 is mounted is made to turn in the opposite 

 sense to the original rotation of the rod, 

 the slight friction between the rod and its 

 axle will gradually tend to stop the mo- 

 tion. But when the rod is nearly at rest 

 close to its highest position one of two 

 things will happen ; either the rod will be- 

 gin to oscillate as before, or after the first 

 oscillation it will be carried past the high- 

 est point and begin to rotate in the same 

 sense as the axle with increasing average 

 velocity. 



On the analytical side we have to notice 

 mainly one point. Before the critical 

 stage the angular motion is expressed by 

 an angle which increases continuously with 

 the time. As the motion gets slower the 

 variations from uniform increase become 

 more and more marked. After the criti- 

 cal stage is passed and the rod is oscillating 

 to and fro, the angle itself varies between 

 two limits which are less than 360° apart, 

 finally settling down to a constant value. 

 If, however, the rod begins to make com- 

 plete revolutions in the opposite direction, 

 the angle diminishes continuously with the 

 time, and its variations from uniformity 



