100 Analytical Considerations respecting the Velocity of Sound. 

 the resulting equation for determiningy will be 



SE + iz+w/ 



(C.) 



*l (#*_#/ df a/ d*/ dp <p/\ 



a? \dx 2 dx* dx dy dxdy dy % dy 2 / 

 Here it is to be remarked, that if <p should contain t, J"is 

 not independent of t, and our result is at variance with the 

 hypothesis by which udx + vdy + wdz was rendered integrable. 

 To ascertain the actual value of <p , the equation (B.) must 

 be integrated, which may be done by successive approximation. 

 I have already shown (Phil. Mag. for April, p. 279), by- 

 taking account of the three first terms of (B.), which are the 

 principal terms, that the only integral applicable to the pre- 

 sent inquiry is ~ 



<p=.mcos — (z—a!t). 



Setting out with this for a first approximation, it may be 

 readily shown that the equation (B.) is satisfied by a series of 

 the form 



tp = m l cos — (z — a't) +w 2 cos — (z — at) +m s cos — {z — a7)-f &c. 



Hence it is clear that the value of z — a'L which causes -— to 



dz 



vanish, gives a value of <p independent of z and t. Thus the 



hypothesis of non-divergent waves is entirely consistent with 



the exact hydrodynamical equations. 



It is worthy of remark, that the above series accounts for 

 the harmonics which are observed to accompany loud notes. 



Another criterion may be applied to the hypothesis of non- 

 divergent waves. Let V be the velocity at a certain point of 

 the fluid at the time £, and V be the velocity at the time t + U 

 at a point of the fluid in advance of the other by h in the di- 

 rection of the line of motion. Then 



Y'^Y+^U + ^h. 



Hence if V' = V, we have 



rfV_ rfV 8s 



dt ds U 



8s . 

 Here — is the rate at which a given velocity is propagated 



through space ; and if this be a constant rate a 1} we obtain 

 the partial differential equation 



dV dV 



-w +a >-dr= > 



