IN CYLINDRICAL TUBES. 247 



This condition is independent of t, and consequently at all points 



distant from the stopped end, any multiple of -, the motion will be 



the same as at that extremity, i.e. it will always equal 0, and there 

 will be perfect nodes at those points. 



16. We may take the general case, and let 



f\at-{il-x)\-^ {a/-(2/ + c-ar)} =j(; {at-{<il->rc, -x)}, 

 and :.v=f{flt — x) — x\a't—{^l^-Cx — x)\, 



^ being still small. The forms of J" and x// being known, that of ^ 

 will be determined ; its period will also be the same as that of J" 

 and ■^. It expresses the velocity of each particle produced by the 

 whole wave actually reflected from B. The nodes will in this case 

 be points of minimum vibration, and not of perfect rest. 



For the sake of clearness we will assume that y(x), and >//(x), are 

 such that 



and therefore 



x(-»)=-x(x), 



that y(»), and ^(z), {and therefore x(*)} admit of only one maximum 



value between x = 0, and 8;=-; and that the ratio which y(s!) bears 



to ylr (%) is always considerable, as by hypothesis it is when those 

 functions have their maximum values. There can be little doubt but 

 that these assumptions are at least approximately true in all practical 

 cases ; and appear as simple as any we can make (and some must 

 be made), in order to give distinctness to our inferences as to the 

 positions of these points of minimum vibration. 



17. For the determination of c, in terms of c, let the origin of 

 t and X be so taken that y(0) = 0, then making at- {2l — x) — 0, 

 we have 



-^(-c) = x(-c,); 



or =\l/{ — c). 



112 



