on the principles of Hydrodynamics, 361 



Retaining now the term involving h^ in equation (a.), this 

 equation gives by integration a certain form of F, which, if it 

 be compatible with vibratory motion, is the particular form 

 that satisfies the condition of constant and identical propaga- 

 tion of all parts of the same wave. A first integral of the 

 equation is readily obtained ; but the complete integration can 

 probably be effected only by successive approximations. To 

 the first approximation we have 



-X2 +-T— o-F=0. 



dv^ 



a^—c^ 



Whence 



17 ( ^^ \ 



i'=OTC0sl , +c| 



consequently, putting 



2-K c b 



— for 



we obtain 



and 



f=:F{v) = m cos -^( IS— ai<\y 1+ ^ -\-c]. 



This is the first approximate value of <p already obtained by 

 the integration of the partial differential equation (B.). By 

 continuing the solution of equation («.) to the third approxi- 

 mation, I find precisely the same expression for <p as that 

 above given for the solution of equation (B.) to the same 

 degree of approximation. We may hence conclude that the 

 solution of (B.) carried to an unlimited number of terms, has 

 the property of satisfying exactly the condition of uniform and 

 identical rate of propagation of all the parts of a wave. 1 have 

 proved this proposition in another way in the Philosophical 

 Magazine for December 1848, p. 465. 



The next step is to introduce into the equation (A.) the 

 condition which it has just been shown that the function <p 

 must satisfy, viz. the condition expressed analytically by the 

 equation 



When by means of this equation the differential coefficients 

 of the second order are eliminated from (A.), the result is 



