IN CYLINDRICAL TUBES. 248 



the possibility of maintaining sonorous vibrations of which the period 

 is - , in stopped tubes of which the length differs considerably from 



?«' . - , particularly if the greatest value of V/ should not be very 



small. If the supposition we have made respecting the continuity of 

 the function (p more particularly, should not be quite true, it is not 

 likely in those practical cases to which we can best refer, to be so 

 far wrong as to render the above reasoning otherwise than at least 

 approximately true. 



12. Our supposition has been that the intensity of the distvu-bance 

 denoted by v//, is considerably less than that indicated by (p, the tube 

 being stopped with some substance having a certain degree of elas- 

 ticity ; if the tube be open, it seems probable from certain pheno- 

 mena, that the reverse of this supposition is true. 



Assuming this to be the case, the expansion of the expression 

 for V may be put under a more convenient form. 



Let 



y{r {at-{2l+ c-x)} =2f{at-(2l- x)] - f, {at- (21+ c' -x)], 



Then 



v=f(at-x)+f{at-{2l-x)}-xj.,{at-(2l + c'-x)} (a), 



and equation (3) becomes 



/(aT + 2l-x)= -/(aT-x) + x|/, {aT-(x + c')} +<p(aT + 2l-x) (4). 



By proceeding exactly as in the former case, we obtain 



v = {-irif(aT-x)+flar-(2l-x)}} 



-fAci{'r+^)-(2l + c'-x)} 

 + 2,^,(-l)-{<^[a(T + ^) -X] + 0[« (t + ^y(2l-x)]} 



1 



ME)(1)- 



