maintained by Forces of Doable Frequency. 509 



.'Wr 

 magnitude of A cannot be greater than ^rYz{ a -~P^)- 



Increase of p 2 —n 2 above the value required to give this 

 maximum value of A renders A imaginary. Hence a 

 collapse o£ maintenance occurs. 



(3) Substitute for -~^j («— pk') the symbol A m denoting 



op rC 



maximum maintained amplitude. 

 Then 



f£A 2 -(p 2 -n 2 )} 2 =a 2 - lpV + (a- p k)£- 



At the excess-tension end, A/A m is very small and if 

 neglected the graph reduces to the form 



}£A 2 = V" 2 -p 2 k' 2 + (p 2 -n 2 ) ; 



and at the defect-tension end, A/A,„. reduces to unity and 

 the graph to the form 



i/3A 2 =p 2 -n 2 . 



The two limiting forms differ only by a constant. This 

 shows that the shape of the graph at the two extremes is 

 similar. 



Therefore A m increases if a increases in value. 



(5) The leastvalueof anecessary formaintenanceisthatwhich 

 makes the expression for the maximum maintained amplitude 



just equal to zero. Since A m = ~-g, {a—pk'), the lower limit 



of the value of a is pk'. Or since a 2 — p 2 \_k' ± 8/37rpkA~] 2 is 

 equal to a square, the least value of u is p(k' + 8/3tt pkA) + 

 and since there is no maintenance, A = and the least value 

 of a is pk'. 



(6) At the excess-tension end, \j a 2 — p 2 k' 2 + (p 2 — n 2 ) is 

 equal to J/3A 2 and p 2 — n 2 is negative. So p' 2 — n 2 cannot 



be arithmetically greater than ^x 2 —p 2 k' 2 . This sets an 

 upper limit to the value of excess tension. 



(7) This upper limit being \/a 2 —p 2 k' 2 increases with 

 increase in the value of a. 



If the frictional term involving the square of the velocity 



