ACTED UPON BY THE VIBRATIONS OF AN ELASTIC MEDIUM. 345 



from the reasoning in the Cambridge Philosophical Transactions, Vol. V. 



Part ii. p. 200. By equating the factor in brackets to 1— n<>, it will 



2 

 be seen that »= - — r, which when S is very small is nearly equal to 2*. 



1 + o 



Before proceeding to a second approximation let us determine the 

 value of the constant h. This will be obtained by finding the value 



f Cm*" "V tr il^ 7* 



of -. , -■ y—j-z> when £ = 0. Now from (11), neglecting the term 



it Tit itt it' fw it t 



involving - as a factor, and differentia ting s 



at 



(' 



d s x _ g dx \ — l ka?m§ 

 W ~~ T"dt *T+7 ~ ^(1 + ty 



and from the same equation, 



d 2 x _ r cPx gx 1 — 8 r e^r kamh - <£ 



df ad? T'l+S ~ a{l + 8)"dF + r(T+7j * '' 



Hence, supposing x = h and j- =0 when t = 0, we readily obtain, 



Whence, 



kam gh 1 — 5 kam$ kaml 



r = ~ T * 1+7 + r(l+5) 2 + RT+1) 



kam _ gh n *s 



d 3 x 



Substituting now this value in the approximate expression for -r— , 



we get 



d*x_ g dx l-S ghat °J- 



d? -~l'dI'TT$ + ~iT- {1 - S)e ' 



* I have already obtained this result in the London and Edinburgh Philosophical Mag- 

 azine for September 1833 (p. 186), in the Cambridge Philosophical Transactions as above 

 cited, and more recently in the Philosophical Magazine for December 1840 (p. 46l). The 

 reasoning in the last of these solutions, not embracing those terms involving the square of 

 the velocity which may be of equal magnitude with terms retained, cannot be considered so 

 complete as that I have now given. 



