.(E)(2). 



244 Mr HOPKINS ON AERIAL VIBRATIONS 



Similarly, we find 

 «*=(-l)"{/(ar-^)-/[«T-(2/-a;)]} >^ 



+ ^l,,{a[T +—j-{2l■^■c-x)} 



+ 2,^,(-l)»-{<^[« (t + ?^) -.V] - 0[« {'r+~) - (2/ - x)\\. ^ 



Reasoning on the expression for v, exactly similar to that used 

 above, will in this case show that sonorous vibrations cannot be 



maintained if / be too nearly equal to an odd multiple of - ; but 



that they can be continued, if / do not differ too much from an 



even multiple of - .* 



13. If we examine the expressions for as in the last article, and 

 in Art. 7, it will appear that the condensations and rarefactions at 

 the surface of the vibrating plate within the tube, are such as to 

 produce forces opposing more strongly the motion of the plate as 

 the lengths of the tubes approximate respectively to those particular 

 lengths for which it will be impossible to maintain the vibrations in 



the tube ; and when the lengths differ from the above by - , these 



condensations and rarefactions are such as to promote the motion of 

 the plate, instead of opposing it. 



14. The expanded expression for v may be put also under another 

 form, which it may be useful to point out for the case in which 

 the intensity of the disturbance denoted by \//, is considerably greater 

 than that denoted by <^. 



* The quantity c' in these general inferences is not taken into account. Its value 

 however is considerable, as will be seen hereafter. 



