IN CYLINDRICAL TUBES. 249 



and in the former 



or l-^ = m\ + ^^', 



consequently, at any point in the tube whose distance from B" = m .-^, 



the velocity will be the same as at B'. These then will be points of 

 minimum vibration in this hypothetical case, and therefore also, from 

 what precedes, in the actual case. 



Making c = 0, we have l—x = m.-, which will give the positions 

 of the nodes when there is no retardation. 



Hence we have this general conclusion with respect to the stopped 

 tube — that if there be a retardation in the phase of the vibration of 

 the extreme section, the positions of the points of minimum vibration 



will all be further from the stopped end by —j, than if there were 



no such retardation, the distances between these points respectively 

 remaining unaltered. 



19. We will now consider the case of the open tube, in which 

 we suppose >|/(a!) to be always considerably larger than J'{%). Assume, 

 as in Art. (12), 



yl,{at-{2l + c-x)}-y{at-{^l-x)}~^, [at - {21 + c' - x)} (8), 



v=f{at-x)-¥f{at-{2l-x)}-^,{at-{2l + c'-x)}. 



First neglecting the function >|/, , v will = whenever 



f{at-x)^-/\at-{2l-x)}; ^ 

 i. e. whenever 



at—x = at—{2l—x) + m'.- {m' an odd number), 

 or I— x = m .-, 



