PROFESSOR KELLAND ON THE THEORY OF WAVES. 541 



c 



,.2 „ (e". ^> g-«^ _ c 6 « 1 + sina (e" ^" + ^"^ T 



c2 a= ff — j r + 6 c a ^ — , .-^ . 



If ^ be the semi-elevation, e=— (^ " '- «""' ') 



C6 



(fail — ea, — ; ;l = 7 — i ; 



o2_^ e -e 



c' = — 



— 7 -i- [1 — e a — J 7 ) 



43. Thus we have obtained the velocity of transmission in a very simple 

 form. As we have before pointed out, the function which properly represents the 

 velocity of vibration is discontinuous ; the value of this function at points of the 

 fluid which are in actual motion cannot be correct, unless it lead to the conclusion 

 that the motion ceases as soon as the wave has traversed a space equal to its 

 length. This conclusion amounts to the fact which it is incumbent on us to 

 prove, that all the superadded fluid, and no more, will pass onwards in the time 

 occupied by the vibration. 



Let Q, be the volume of superadded fluid ; R the volume of the portion which 

 is carried forwards during the time of vibration. 



c a 



Q=/ \ dx—(e'''~e-''')(l + sme) 

 ^ — Co, 



4 



gKZ _. g« /t + a m + « m sin ^ 



= ±fdxl e"-^ ~ .— ^ +^- (."'' + .-'^•").+ ... 

 c a, \ 2 



c a\ 2 J 



VOL. XIV. PART II. 5 A 



