(A.) 



on the principles of' Hydrodynamics. 357 



^ \dx^ dx^ doc dy dxdy dy^ dy^ / 



J dz'dzdt ^ dz^' dz^ 



In treating this equation, it may be assumed, since/* and <p 

 contain no variables in common, that if particular and con- 

 sistent values be substituted fory, x and y, the resulting equa- 

 tion is true for general values of (p, z and t. Now since, by 

 hypothesis, the motion is vibratory, the function /must have 

 a maximum value. This value may be assumed to be unity, 

 becauseymay be regarded as a numerical quantity, and one 

 value of it may be taken arbitrarily. Let therefore the values 

 of .r and^ given by the equations 



dx ^ dy ^ 

 satisfy the equations ^ 



•^~ ' dx^ "^ %2- ^2» 



the negative sign in the latter being a consequence of the sup- 

 position of a maximum. Then by substitution in (A.) we 

 obtain the following equation for determining <p : 





dz^ dt^ dz dzdt dz^ dz^' 



(B.) 



It does not appear that this equation is generally integrable: 

 but an integral applicable to the present inquiry may be ob- 

 tained by successive approximations, the terms involving the 

 first power of <p being first considered. The arbitrary func- 

 tion of the time may be supposed to be included in <p. Thus 

 to the first approximation we have 



The integral of this equation, as I have shown in the Phi- 

 losophical Magazine for April 1848 (p. 278), may be obtained 

 in an infinite series involving two arbitrary functions. In 

 investigating the velocity of propagation, one of the arbitrary 

 functions must be made to vanish. This having been done, 

 and the following substitutions made, viz. 



