Hence 



• (45) 



Source on the A.ris of a Cylindrical Tube. 663 



The part oF the emission due to the pth. term of the sum 

 in (-43) is 



| W {J (A,a)}» " ""T^VTTa^ 



I 6' 2 + 6 P j 



or, writing fi/ke = v, e p Jk = x, 



V2'87ra 2 {Jo(V)^' {v 2 + ^ 4 }* 



.... (46) 



As the forced period tends to coincidence with a free period, 

 v being supposed to have a definite value, this expression 

 tends to the limit 



P r. 



V2'87ra 2 {J (£a)} 2, 



whichever of the two periods was originally the greater. 

 Thus the discontinuity which occurred in the case of no 

 friction does not now present itself. 



This problem is interesting as furnishing an instance of a 

 "double limit." If in the expression (46) we make first oo and 

 then v tend to zero, that is to say, if we make the periods 

 tend to coincidence and then make the friction vanish, the 

 expression tends to an infinite limit. If, however, we make 

 v vanish before a?, the limit to which the expression tends is 

 zero or infinite, according as the upper or lower sign is taken, 

 that is according as the forced or free period is originally 

 the greater. 



The accompanying diagram shows the variation of the last 

 factor in the expression (46) as a function of x 2 . In the 

 lower of the two curves the value of v has been taken to be 

 1/100, and in the higher 1/400. The left-hand side of the 

 figure corresponds to the upper sign in (46), that is to say, 

 to the case when the forced period is slightly greater than 



2X2 



