444 Mr. stokes, ON THE THEORY OF OSCILLATORY WAVES. 



2. In proceeding to a first approximation we have the equations (2), (3) and the equation 

 obtained by omitting the small terms in (10), namely, 



s-^ - c^— ^ = 0, when « = (ll). 



The general integral of (2) is 



the sign 2 extending to all values of A, m and »i, real or imaginary, for which m? + n- =0: 

 tile particular values of <p, Cx + C', Dy + D', corresponding respectively to n = 0, to = 0, must 

 also be included, but the constants C', D' may be omitted. In the present case, the expression 

 for must not contain real exponentials in x, since a term containing such an exponential would 

 become infinite either for x= - 05 , or for x= + eo, as well as its differential coefficients which 

 would appear in the expressions for u and v ; so that m must be wholly imaginary. Replacing 

 then the exponentials in w by circular functions, we shall have for the part of (p corresponding 

 to any one value of to, 



(^e""' + A'e-""-') sin nuv + (Be'"!' + B'e-"") cos m.v, 



and the complete value of (p will be found by taking the sum of all possible particular values of 

 the above form and of the particular value Cx + Dy. When the value so formed is substituted 

 in (3), which has to hold good for all values of x, the coefficients of the several sines and cosines, 

 and the constant term must be separately equated to zero. We have therefore 



D=0, A' = e-'"''A, B' = €'""'B; 



so that if we change the constants we shall have 



<^ = Ca; + S (e""*-"' + e""'"-") {A sin mx + B cos ma), ... (12), 



the sign 2 extending to all real values of w, A and B, of which to may be supposed positive. 



3. To the term Cx in (12) corresponds a uniform velocity parallel to x, which may be supposed 

 to be impressed on the fluid in addition to its other motions. If the velocity of propagation be 

 defined merely as the velocity with which the wave form is propagated, it is evident that the 

 velocity of propagation is perfectly arbitrary. For, for a given state of relative motion of the 

 parts of the fluid, the velocity of propagation, as so defined, can be altered by altering the value 

 of C. And in proceeding to the higher orders of approximation it becomes a question what 

 we shall define the velocity of propagation to be. Thus, we might define it to be the velocity 

 with which the wave form is propagated when the mean horizontal velocity of a j>article in the 

 upper surface is zero, or the velocity of propagation of the wave form when the mean horizontal 

 velocity of a particle at the bottom is zero, or in various other ways. The following two definitions 

 appear chiefly to deserve attention. 



First, we may define the velocity of propagation to be the velocity with which the wave form 

 is propagated in space, when the mean horizontal velocity at each point of space occupied by the 

 fluid is zero. The term mean here refers to the variation of the time. This is the definition 

 which it will be most convenient to employ in the investigation. I shall accordingly suppose 

 C = in (12), and c will represent the velocity of propagation according to the above definition. 



Secondly, we may define the velocity of propagation to be the velocity of propagation of the 

 wave form in space, when the mean horizontal velocity of the mass of fluid comprised between 

 two very distant planes perpendicular to the axis of x is zero. The mean horizontal velocity of 

 the mass means here the same thing as the horizontal velocity of its centre of gravity. This 

 appears to be the most natural definition of the velocity of propagation, since in the case considered 

 tliere is no current in the mass of fluid, taken as a whole. I shall denote the velocity of propaga- 

 tion according to this definition by c. In the most important case to consider, namely, that in 



