208 WAVES IN AIR. [CHAP, vm. 



&amp;lt; = ^ (x, y, *), ~-= x (x,y,z) ............ (25). 



The mean initial values of these quantities over a sphere of radius 

 r described about (x, y, z) as centre are 



&amp;lt;j&amp;gt; = I l^jr (x + Ir, y -f mr, z + nr) dvr, 



-jfc = J^.// % ( x + lr, y + mr, z + nr} dvr, 



where Z, m, n denote the direction-cosines of any radius of this 

 sphere, and d^ the corresponding elementary solid angle. Com 

 paring with Art. 168, we see that the value of &amp;lt;f&amp;gt; at any subsequent 

 time t is 



1 1 x ( x + 



y + mct &amp;gt; z + nctyd-a- ...... (26). 



171. We have so far assumed the velocity and the conden 

 sation to be so small that their squares and products may be 

 neglected. The results obtained on this supposition are indeed 

 sufficiently accurate for most purposes ; but it is worth while to 

 notice briefly the solutions of the exact equations of motion of 

 plane waves which have been obtained, independently and by 

 different methods by Earnshaw* and Riemannf. 



Riemanns Method. 



Riemann starts from the ordinary Eulerian form of the equa 

 tions of motion and continuity, which may be written 



du du_ dpdlogp 



dt^ U dx~ dp dx ^ 



dt dx dx 



} 



* Phil. Trans. 1860. 



t Gott. Abh. c. 8, 1860. Reprinted in Werke, p. 145. 



