Prof. Challis on the Principles of Hydrodynamics. 89 



«(/i— 1) of a* satisfies to the same degree of approximation the 

 principal terms of the equation 4=0, and consequently that 



71— — approximates to a root of that equation in proportion as 



13 



n is large, So n+ ^, n + -^, &c. are roots, n being indefinitely 



great. Thus the infinite roots of -^ = form an arithmetic 



series of which the common difference is unity, and are means be- 

 tween the infinite roots of /=0. If, therefore, r^, rc^ be the radii of 

 two consecutive cylindrical surfaces in which the condensation = 0, 

 or the transverse velocity = 0, we shall have \^e{r^—r{) = 1 ; 

 and putting D for r^—ri. 



Now suppose a series of vibrations such that the condensation 



277- 



and velocity each varies as the function 0, or m sin — - [z—dt + c), 



A. 



to be propagated along the rectilinear axis of the motion in the 

 positive direction, and an exactly equal series to be propagated 

 in the opposite direction, the possibility of the coexistence of two 

 such series having been previously proved. The effect of the two 

 series will be to form along the axis at equal intervals points of 

 no condensation, and exactly intermediate to these, points of 

 quiescence. Thus the motion along the axis will be similar to 

 the transverse motion which we have just considered. Also the 



general expression for the transverse velocity, viz. ^ -j-, proves 



that the transverse vibrations to which a single series gives rise 

 will be executed in the same time as the direct vibrations along 

 the axis. This will clearly be the case also when there are two 

 opposite and equal series. Hence the interval between two con- 

 secutive points of quiescence on the axis must be equal to the 

 interval between two consecutive surfaces of no transverse velo- 

 city. The former interval is ^, and the latter has been shown 



1 



above to be — -=. Hence 



ve 



1 ^\ 



But it has been proved that the velocity of propagation is 



Phil. Mag.B. 4. Vol. 5. No. 30. Feb. 1853. H 



