242 Mh HOPKINS ON AERIAL VIBRATIONS 



In this case it is manifest that the consecutive terms taken in the 

 order just mentioned will destroy each other ; and there will con- 

 sequently be no accumulation of motion in the tube, and the 

 vibrations will go on uniformly. Again, let 



2l = m\, or / = 2m. -. 



4 



In this case the values of the successive terms taken as before in 

 the expression for v will be equal, and with the same sign. Hence, 

 if we take x of any value, except such as would render 



<(>{at-x) = <p {at-{9.l-x)], 



f which value of x is I — m -\ , it is manifest (since the value of (p 



is greater than that of \|/), that the motion will constantly increase 



for such points, and will soon become greater than is consistent 



with our original suppositions. Such a vibration then cannot be 

 maintained. . 



11. Again suppose the functions (p, f, and ^, to be continuous, 

 and suppose 



2/=m'^+2\', or / = m'^+\', 

 2 4 



X' being any quantity less than -; the consecutive teims of 1.(f>(%), 



tit 



will not then destroy each other, but as the number of pairs of terms 

 increases, the sum will increase till ^(s; + 2r/) becomes negative, it will 

 then decrease, after having thus attained a maximum value. Maxima 

 and minima values will thus occur alternately, and the same will hold 

 for 2. >//(»). If these maxima values do not render v greater than our 

 original suppositions allow, the vibrations may be maintained. 



Since these maxima values are 0, when l = m'.-, and greatest 



when l=m' .-, we conclude that they will be intermediate for inter- 

 mediate values of I, following some continuous law. Hence we infer 



