Mr. stokes, on THE THEORY OF OSCILLATORY WAVES. 449 



at variance with the results obtained by Mr. Airy for the case of long waves*. On proceeding 



to a second approximation, Mr. Airy finds that the form of long waves alters as they proceed, 



and that the expression for the velocity of propagation contains a term depending on the height 



of the waves. But a little attention will remove this apparent discrepancy. If we suppose 



mh very small in (19), and expand, retaining only the most important terms, we shall find for 



the equation to the surface 



3 ot^ 

 y = acosmx- — ;— -cos2m*. 

 im-h 



Now, in order that the method of approximation adopted may be legitimate, it is necessary that 

 the coefficient of cos 9.inx in this equation be small compared with a. Hence , and therefore 



— — must be small, and therefore - must be small compared with ( — ] . But the investigation 

 A ' h V X / 



of Mr. Airy is applicable to the case in which — is very large; so that in that investigation 



- is large compared with (-) . Thus the difference in the results obtained corresponds to a 

 difference in the physical circumstances of the motion. 



12. There is no difficulty in proceeding to the higher orders of approximation, except what 

 arises from the length of the formulae. In the particular case in which the depth is considered 

 infinite, the formulae are very much simpler than in the general case. I shall proceed to the third 

 order in the case of an infinite depth, so as to find in that case the most important term, depending 

 on the height of the waves, in the expression for the velocity of propagation. 



For this purpose it will be necessary to retain the terms of the third order in the expansion 

 of (7). Expanding this equation according to powers of y, and neglecting terms of the fourth, &c. 

 orders, we have 



g(p- c'(p + (g(p,- c-<p, )y + (g-0^ , - c=(^,;') ^ + 2c ((p'fp" + (p, (p,') 



+ '2ci<p'(p"+<p'(p''+(p^^<p^'+(j)^(p^^')y - <p''(p"-2<p'<p^(p'- (p^'(p^^= (25). 



In the small terms of this equation we must put for (p and y their values given by (21) and (22) 



re.spectively. Now since the value of to a second approximation is the same as its value to 



a first approximation, the equation g(j),- c^(p" = is satisfied to terms of the second order. But 



y' 

 the coefficients of y and — , in tlie first line of (2.'5), are derived from the left-hand member of 



the preceding equation by inserting the factor e'"'", differentiating either once or twice with 



respect to y, and then putting y = 0. Consequently these coefficients contain no terms of the 



second order, and therefore the terms involving y in the first line of (25) are to be neglected. 



d 

 Tlie next two terms are together equal to c — (0 - + (pf). But 



(p'"- + <p^ = m-d'c', 



which does not contain x, so that these two terms disappear. The coefficient of y in the 

 second line of (25) may be derived from the two terms last considered in the manner already 

 indicated, and tlicreforc the terms containing y will tlisappear from (25). The only small terms 



• Encj/olopadia Melropnlituim, Tiiks niirl ll^avcs, Articles lilD, &c. 



3m 2 



