198 DYNAMICAL THEOEY OF SOUND 



u l} u 2 at P and w/, u% at P'. The as-component of the 

 acceleration of this particle will be the limit of 



The limit of the first term on the right is du/dt, the rate of 

 change of u at P. Again u% u z is the difference of simul- 

 taneous velocities at the points P, P', so that, ultimately, 



*,), ............ (2) 



where du/ds is a space-differentiation in the direction PP', and 

 q is the resultant velocity \/(u 2 + v* + w z ). The final expression 

 for the acceleration parallel to x is therefore 



du du , 



Similar values are obtained in like manner for the other 

 components. If (I, m, n) be the direction-cosines of PP', 

 we have 



du du dx du dy du dz 

 ds dx ds dy ds dz ds 



j du du du 

 = t^+ra + 7i , (4) 



dx dy dz 



Philst u=lq, v = mq, w = nq (5) 



!ence we may write (3) in the form 

 du du du du ._. 



dt* U dx +V ty+ W 3z> 



which is familiar to students of Hydrodynamics. 



It has been thought worth while, as a matter of principle, 

 to accentuate the changed point of view, but in the application 

 to motions which are treated as infinitely slow the distinction 

 loses its importance. The second term in (3) is then of the 

 second order in the velocities, and the component particle- 

 accelerations may be identified with du/dt, dv/dt } dw/dt. The 

 extent of the error here involved, in acoustical questions, may 

 be estimated as in 60 by a reference to plane waves of sound. 

 If 



= a cos k (ct x), (7) 



