Source on the Axis of a Cylindrical Tube. 659 



greater than thepth free period, the energy expression is 



«7ra 2 i 87^aV s {J ^a)}•' , 



and is finite. Thus the emission of energy is discontinuous 

 for values of the forced period coinciding with a free period ; 

 viz., it tends to a finite or infinite limit according as the critical 

 period is approached from a higher or a lower value. 



If we had started on the supposition that the period of the 

 source is coincident with one of the free periods, say k=k p , 

 we should have found 



*=2^?Bin*(c«-,)+ {A- 2^3^ } JoWoosto 



^ JoC^Tpsin (kct—u s z) ^ J {k s r)e" m ^ cos kct ^ 

 + f 27raV s {J (^«)} 2 P +i27ra 2 m s {J {k s a)Y 2 ' 



where A is arbitrary. The rate of emission of energy is 

 therefore 



hirci 2 i 87ra 2 /jL s {J (k s a)} 2 ' 



which is the same as the limit to which the emission tends as 

 the forced period approaches coincidence with the pih free 

 period from a slightly greater value. 



The reason why the emission of energy is very great when 

 the forced period is slightly less than the free period, but 

 not when it is slightly greater, is found in the difference in 

 the relationship between the phases of the pressure and 

 velocity in the two cases. When the forced period is slightly 

 the smaller, the very large term in the expression for the 

 pressure is in the same phase with the corresponding term 

 in the velocity, and is therefore in a condition favourable 

 for doing work. But as k increases through a root of 

 J u \k s a) = 0, the important term in the expression for 'dcfi/'dt 

 undergoes a change of phase of a quarter period, and thus 

 differs in phase by this amount from the corresponding 

 velocity. Thus it does no work on the whole. 



It is obvious that in any uctual case the discontinuities 

 noticed above cannot occur ; and the problem therefore needs 

 some modification in the case of approximate isochronism. 

 This is effected by the introduction of dissipative forces of the 

 nature of friction, viscosity, or heat conduction, which have 

 hitherto been neglected. They are in fact, as a rule, unim- 

 portant except ia the critical cases. The simplest hypothesis 



