Mr. stokes, on THE THEORY OF OSCILLATORY WAVES. 445 



which tlie depth is infinite, it is easy to see that c - c, whatever be the order of approximation. 

 For when the depth becomes infinite, the velocity of the centre of gravity of the mass comprised 

 between any two planes parallel to the plane yz vanishes, provided the expression for u contain 

 no constant term. 



4. We must now substitute in (11) the value of 0. 



(^ = 2 (e-C-^* + j-'-C'-i/)-) (J sin moo + Bcosmx) (13) ; 



but since (11) has to hold good for all values of ,v, the coefficients of the several sines and cosines 

 must be separately equal to zero: at least this must be true, provided the series contained in (ll) 

 are convergent. The coefficients will vanish for any one value of m, provided 



^e-*-e-* 



Putting for shortness 2mh 



(14). 



which is positive or negative, ^i being supposed positive, according as 



V 1 .2.3 I 



and is therefore necessarily negative. Hence the value of c given by (14) decreases as fx or m 

 increases, and therefore (11) cannot be satisfied, for a given value of c, by more than one positive 

 value of m. Hence the expression for <p must contain only one value of in. Either of the terms 

 A cos mx, B sin ma: may be got rid of by altering the origin of .r. We may therefore take, for 

 the most general value of (p, 



.^ = ^(e'"'*-J" + £-""*-*') sin mx (15). 



Substituting in (8), we have for the ordinate of the surface 



niAc , , ■ ^ ,.^ 



y = (6j + e'" ) cos mx (10), 



k being = 0, since the mean value of y must be zero. Thus everything is known in the result 

 except A and m, which are arbitrary. 



5. It appears from the above, that of all waves for which the motion is in two dimensions, 

 which are propagated in a fluid of uniform depth, and which are such as could be propagated into 

 fluid previously at rest, so that udx + vdy is an exact differential, there is only one particular kind, 

 namely, that just considered, which possesses the property of being propagated with a constant 

 velocity, and without change of form ; so that a solitary wave cannot be propagated in this manner. 

 Thus the degradation in the height of such waves, which Mr. Russell observed, is not to be 

 attributed wholly, (nor I believe chiefly,) to the imperfect fluidity of the fluid, and its adhesion to 

 the sides and bottom of the canal, but it is an essential characteristic of a solitary wave. It is true 

 that this conclusion depends on an investigation which applies strictly to indefinitely small motions 

 only : but if it were true in general that a solitary wave could be propagated uniformly, without 

 degradation, it would be true in the limiting case of indefinitely small motions ; and to disprove 

 a general proj)osition it is sufficient to disprove a particular case. 



6. In proceeding to a second approximation we must substitute the first ajjproximate value of 

 0, given by (15), in the small terms of (lO). Observing that A; = to a first approximation, and 

 eliminating g from the small terms by means of (14), we find 



g(f>, - c'(ji"- 6A'm'c sin 2mx = (17). 



