128 PROFESSOR KELLAND ON THE THEORY OF WAVES. 



... (4) 



the origin being placed at the quiescent surface of the fluid, so that z shall be a 

 very small quantity. 



The symbol 2 is such that it expresses the sum of the two functions f(m, t\ 

 f,(m, t), the latter being multiplied by sin mx; and it evidently requires that, 

 when it stands before sin m x the quantity not expressed to which it applies, it is 

 to have the negative sign. 



Also equation (3) is 



y^co ' 1 



_./ - / A / m z wi z-J-2 n\ j 



g 2 / cos m xf(m, f)m(e e ) dm 



Jo 



r*> d 2 f(m, t) . mz , mz+Zh-. , n 



+ 27 coswta; ^5 - (e +e )dm=0. 



Jo dl 2 



We can satisfy this equation by making 

 <Pf(m,i) 



m z mz+2 h 

 -re 



Denoting gn? m j . ^ by c m , (5) 



The solution of these equations evidently is 



f(m, t) = <p (m) cos c m t+ -^ (m) sin c m t 



f, (m, f) = <p, (m) cos c m t + ^ (m) sin c m t, 

 which being substituted in the value of <p reduces it to 



<t> = -S.fcosmx cos c m t (^ + e-^ h } $(m)dnt 



Jo 



+ 2 f " cos m x sin c t (e my + e - Jai y+^ h ') ^ () dm. 



Jo 



Now one condition is, that when x=k, u=Q for all values of y, this gives 



sin mk(p (m) + cos mk(p t (m) = 



sin m k 4 1 (m) + cos m k -^ (m) = 



(m) dm 



~ m y+2 A ) tan m k $ (m) d m + &c. 



=/ cosct(e m y + e- m y+ 2 ' 1 } -^^- cosm(x-k\ 

 Jo cos m K 



+ &C. 



