Direction, of Sound through Air. 129 



Hence, eliminating n between these last, we shall get 

 m = funct. (©^j), 

 .\/ 2 =/ 2 (ma) 2 ) = funct. (o),fi) 2 ); 

 therefore one of the first integrals of (1) will be of the form 



funct (0)^3) = 0, 

 whence it is clear that we must have 



The substitution of this value of v in equations (2) gives us 

 _, . du x a 2 du x 



' u .du x du x 

 from which it follows that we must have 



and fU . « . 



,\ #==.+ « loge «* + Cj 



or, since ?; vanishes when a x = 1, 



v = + a log e a ; 



from which it is clear that the solution obtainable in this manner 

 must be identical with that already deduced. 



2. The only way of escape from this conclusion is to suppose 

 that (16) is illusory, considered as a relation between the partial 

 differential coefficients of f 2 — or, in other words, that we have 



/i'K)-/.'W=o, 



a 



/i'K)=o. 



The last of these implies that f } is a function of co 1} x } and t 

 only ; hence, writing the former 



the auxiliary equations for its integration are 



= ae 2« dt+dxj = dco v 

 which give us 



