Mr. stokes, on THE THEORY OF OSCILLATORY WAVES. 443 



If c be the velocity of propagation, u, v and p will be by hypothesis functions of x - ct and y. 



It follows then from the equations m = — ^ , y = -J- and (I), that the differential coefficients 



da! ay 



of (p with respect to x, y and t will be functions of ,r - ct and y ; and therefore (p itself must 



be of the form f(x-ct, y) +Ct. The last term will introduce a constant into (l) ; and if 



this constant be expressed, we may suppose ^ to be a function of x — ct and y. Denoting x - ct 



by .r', we have 



dp dp dp dp 



dx dx ' dt dx ' 



and similar equations hold good for 0. On making these substitutions in (1) and (4), omitting 

 the accent of a;, and writing — gk for C, we have 



'-'(-" -4! -si(sy-0] <". 



ld&> \ dp deb dp 



(df - '^j i ^ rff rff = °' ^'''^" ^ = ° "• («)■ 



Substituting in (6) the value of p given by (."j), we have 



dy dx'' \dx dx' dy dxdyl 



_ /d<pydr^_^dj> d^ ^ _ fd^yf^^^^ , 



\d,vl dx- dx dy dxdy \dy I dy^ 



when ^(j' + '^)-S-i{(g)^0] = O (S)- 



The equations (7) and (8) are exact ; but if we suppose the motion small, and proceed to the 

 second order only of approximation, we may neglect the last three terms in (7), and we may 



easily eliminate y between (7) and (8). For putting <p', (h , &c. for the values of ~ , — -£- , &c. 



dx dy 



when y = 0, the number of accents above marking the order of the differential coefficient with 



respect to x, and the number below its order with respect to y, and observing that A; is a small 



quantity of the first order at least, we have from (8) 



g (y + /c) + c (0' + (p;y) - 1 ((f)'' + (p;') = 0, 



whence y = - k - - (pi + ~ (p^ {k + - <p') + — ((p'' + <p^).* (9). 



Substituting the first approximate value of y in the first two terms of (7), putting y = in the 

 next two, and reducing, we have 



g<P, - c'<p" - (g(p^, - c^(p") (k+- 0') + 2c (cp'<p"+(p^(p') = 0. ... (10). 



<p will now have to be determined from the general equation (2) with the particular conditions (3) 

 and (10). When is known, y, the ordinate of the surface, will be got from (ij), and k will 

 then be determined by the condition that the mean value of y shall be zero. The value of p, if 

 required, may then be obtained from (.5). 



• The reader will ubHcrve thai the y in this equation is the ordinate ol" the tturlace, whereas the y in {1) and ('2) is the ordinate of 

 any point in the fluid. The context will always show in which Dense y Is employed. 



