222 Mr. Ralph HoliMES on 



Hence 



i[^^Jl\^yW^uf\ (iii.) 



p\dt dx\ ^ydt dxj ^ 



P- 



Hence, eliminating/, p' from (i.), (ii.), (iii.), we obtain 

 d'^u yfi (dhi . I dp dii^ 



yp/d'u idp du\ xld^_x dp 4\ . . 



p-ydx"- p dx dx) p\dx^ p dx dx) ^ '' 



Supposing that the changes in the pressure and density 

 are so small that we may neglect their second differentials 

 and products and powers of differentials above the first, 

 the equation (iv.) to determine ?/ becomes 



py d'^ii y dp du d'u , ^ ^ 



p dx ' p dx dx dt' 



If/ and p were constant, a solution of this equation may 

 be written 



K'vf)-"('v?) 



where A and B are constants. 



Let us therefore assume, as a solution of equation (v.) 



-('■/;b)-"("/;2) 



where A and B are now slowly varying functions of x, such 

 that their second differentials and products of their first 

 differentials with the first differentials of p and p may be 

 neglected. We have, putting q^= ^'', 

 du dA ^B^ ^ _ 



dhi 



^^, = A/" + BF' 



Substituting these values in equation (v.), we have 

 \ q dx q'-dx pq dxy \q dx q^ dx pq dx) 



