188 Mr. Appleton and Dr. B.van der Pol on a Type of 



For all values of B for which B>A 2 (Region 1), we see 

 that the roots are either zero or imaginary, so that no 

 sustained oscillation is possible. In Region 2, however, for 

 which we have A 2 >B > 0, various possibilities arise according 

 to the initial value of a 2 . If the initial amplitude is small 



so that a 2 2 >a s 2 >a 2 , we see from (11) that ^a 2 <0, and thus 



the amplitude will tend to decrease to zero as indicated by 

 the arrows. If, however, the initial amplitude is such that 

 d 



2 > and the amplitude will increase 



a 2 2 >a 2 > a 3 2 , then -j a 



to a 2 . Further, if a 2 >a 2 2 >a 3 2 , then -ya? 



at 



< and the ampli- 



tude again finally reaches a 2 . We thus see that in Region 2 

 no finite stable amplitude will be reached automatically 

 unless by some external agency we succeed in producing in 

 the system an amplitude a such that a 2 >a 2 , in which case 

 the amplitude automatically builds up to the stable value a 2 . 

 For any gradual variation of B after such an amplitude has 

 once built up, the amplitude value a 2 is maintained. 



Fig. 5. 



As a method of testing this theory, we may imagine a 

 cycle of operations in which B is varied from a large 

 positive value to a negative one, and then the process 

 reversed. We see from the above discussion that the cycle 

 will be irreversible and of the type illustrated in fig. 5, 

 where the relation between a 2 and B is shown. The value 



