212 Prof. Challis on Hydrodynamics. 



posing that ~ = and -77=0 where p = l. Letp = l + cr. Then 



to the second order of approximation, which is all that is required 

 for the present purpose, it will be found that 



2ni 2 /c 2 _ m 2 , a ..v . ,„, 



aa-ttuc sm qp- 3a ^_^ cos2gy* + ^ (k 2 - l)sm 2 qf t, (0) 



and, w being the velocity along the axis, to the same approxi- 

 mation, 



w = msinqfi-- 3a ^™^ coa2qp (7) 



As the above expressions for w and <t have been obtained ante- 

 cedently to any arbitrary conditions of the motion, the inferences 

 drawn from them must relate exclusively to circumstances of 

 the motion that are not arbitrary. In the first place, since fi is 

 equal to z — tcat + c, it follows that both the velocity and the 

 condensation are propagated with the uniform velocity ica. As 

 this is true however far the approximation be carried, the law of 

 uniform propagation is independent of the magnitude of the 

 motion, and holds good in every instance of propagated motion. 

 Again, from the equations (/3) and (7) the following equation, 

 to the same approximation, is readily deduced : 



/ o -, ^ wr 



acr = kw + (fc z — 1) — • 



£a 



This result informs us that the condensation corresponding to a 

 positive value of w is greater than the rarefaction corresponding 



to an equal negative value by (/c 2 — 1) —%. The reason for this 



law will be apparent by considering that, as the motion is wholly 

 vibratory, the forward excursion of each particle must be equal 

 to its excursion backward, and that this will be the case if at a 

 given instant the variation of a for a given variation of z be 

 greater at a point of condensation than at the corresponding 

 point of rarefaction in the proportion in which the density is 

 greater at the former point than at the latter. 



In the third place we may conclude that as the forms of the 

 expressions for w and a were obtained antecedently to the sup- 

 position of any arbitrary disturbance, they must be applicable 

 generally to cases of disturbance producing vibratory motions. 

 Also since the equation (a) is exact, and no limitation was made 

 as to the magnitude of the motion in obtaining the particular 

 solution now under consideration, it follows that that solution is 

 applicable to motions of all magnitudes. But in proportion as 

 the motions are larger ; a greater number of terms of the series 



