334 Mr. EARNSHAVV, ON THE MATHEMATICAL THEORY OF 



The equations for ihe pressure are, 



dj.p = - d,u - ud^u = (c - u)djU, 



and dyp = - g + n'y ; 



.•. p = - \(c — uY - gy + \n^y^ + constant. 



Now for a particle in the surface of the fluid p is constant ; and if z be the value of i/ 

 for such a particle, then 



constant = (c - iif + ^gz - n^x' (23). 



But the value oi c - u is known in terms of sc from (l6), and consequently, 



constant = — r-, (« '•' + e '^ \ +2gz - n/'z^ (24) 



is the equation which gives the form of a wave, t is here to be considered constant. 



Again, when z = h, u = 0, 



.•. constant = c' + 2gh - n^hr from (23). 



ch 

 Also when z = k, c - ti = ~ from ig, and consequently, 



constant = -— + Sgk - M-fc' from (23) ; 

 n 



.: = cM 1 - -j + 2g {h - k) - n- {h- - k') ; 



and .-. c' + 7i'kP = ^^^ 



h + k 



And if in this equation we write the value of n from (20), we obtain the following final 

 equation for the velocity of transmission. 



/_2^\ 

 U + k) 





jlog^ tan (^^ + 1 COS- -^) I 



.(25). 



It is to be remarked, that if h be very nearly equal to k, the denominator of the fraction 



on the right-hand side of this equation becomes equal to 1 ; and tiie numerator equal to gk, 



so that c = s/gk in that case; which is the empirical formula used by Mr. Russell. If h be 



o Of- 1£^ ■ fc-^-lf\ 



nuich less than k, then — — ( =gk. ) is greater than gk; but in that case the 



h + k \ h + k) 



denominator is greater than 1, and consequently there is a tendency to compensation which causes 



the value of c to lean sensibly towards the value s/gk ; which accounts for the near agreement 



of Mr. Russell's formula with experiment ; and shews that he was mistaken in imagining the 



velocity of transmission to be entirely independent of the length of a wave. 



