Theoretical Determination of the Velocity of Sound. 34-1 



tains logarithms, and the general solution of the latter contains 

 exponentials and multiples of x and y ; all of which, as the)' 

 may be infinite, are unsuited for our purposes. We can only 

 adopt the valuey= constant ; and the definition of the air- 

 disturbance will be given by equation (S.) reduced to the form 



dt^ "■ dz^ -"' 

 from which the usually received consequences follow. 



(/3.) We may suppose i^ to have a value; and this value 

 must have the sign which Professor Challis has given it, as 

 otherwise tlie solution of equation (4.) would contain expo- 

 nentials. The most general solution of the equation which 1 

 can then give, free from exponentials, is 



/=S.A cos[px + qy-\-B)i 

 where 



Multiplying this expression by Professor Challis's expression 

 for (p, namely, 



7« . cos< {71 )z— (n-\--\at + C 



and expanding the product of the cosines, we find that (he 

 value ofy will be expressed by the two series of terms 



S .-^ . cos < ( n js +7Jar + <7j/ — ( « + - ) at + C' > 



+ 2.'^-cos{(«-i).-p^-<7i/-(n+^)^^ + C"}. 



Each of these represents a series of plane waves of indefinite 

 extent, the position of the plane at any instant being defined 

 by the equation 



+px + qy = a given quantity, 



li = 



I n— — )z—p.v — qy = a given quantity. 



il for the normal drawn from the origii 

 I the former plane, it is easily found th; 



(" ~n)^ '^P^' ^ "^y V ~ M ) ^ '^^'^ "^ ^^' 



If we put 11 for the normal drawn from the origin of co-ordi- 

 nates upon the former plane, it is easily found that 



\/{("- s)'+'''+''} "+ i 



