IN CYLINDRICAL TUBES. 245 



This is deduced, by assuming 



i,,{at-(x + c')}= (if (at -a;)+yl.'{ai-(x + c")}, 

 or, 



x/. {«/- (x + c)} = (2 - /3) /(«^ -x)+ir' {at -(x + c")}. 



Then the equation (a) (Art. 12) becomes 



v=f{at-x) + {l-l3)f{at-{2l-x)}-i,'{at-{2l+c"-x)} (/3). 



We may observe, that since the vibration denoted by \j/, is pro- 

 duced by that denoted by Jl it seems a necessary consequence that 

 their periods must be the same. Their phases also are nearly so ; 

 and if in addition we assume that the Jbrm of the function ex- 

 pressing the one motion, does not differ very widely from that ex- 

 pressing the other, (however the intensity of the vibrations may differ) 

 it is manifest that /3 may be so taken that the intensity of the 

 vibrations denoted by the unknown function \j^' shall be small com- 

 pared with that indicated by <p. 



Equation (4) becomes 



f{ar + 2l-x)=-Cl-ft)f(aT-x) + i.'{aT~(x + c")}+(j>(aT + 2l-x) (5), 



= -hf{ar -x)^-^' {a-r - (^ + c")} + («t + 2/- x), 

 if 1-/3 = *. 



This gives us 



And the equation (/3) becomes, (when t=T-\- j, 



v^{-hY {f{aT-x)+hflar-{2l-x)]} 



+ S,,,(-&)-|>/.'{«[t+ ^^^^^)-^ ]-(x+0}+&^^1«[t+ ^^''^^^-/ ]-(2/+c"-;»^)}I 



-^'{a(r + ^)-{2l + c"-x)]. 



r-n ^ W/ \ (t ) 



Vol. V. Part II. 1 1 



