PROFESSOR KELLAND ON THE THEORY OF WAVES. 509 



Hence the equation A . (g) = ^ . (g) gives 



a c . COS 6 f y — 2a sin cos fy f y + sin 6 cos 6 (Pyf'y + ¥'y f'y) 

 = a?cFy cos e-2a? Fyfy sin d cos 6 + aF'y Fy sin 2d. 



Equating coefficients, we obtain 



fy = o^Fy (2) 



-<-/yfy+ ^--'^'"^^^^^ +u^Fyfy-F'yFya=() 

 or --F'yF"y + \(lFyF"'y + lF'yF"y\ =0 



The complete solution of this equation is 



fyrrzbe'y + b'e-'y 

 F'y = a{^be'y + b'e-'y);Fy = be''y-b'e-'''' (2) 



If we substitute this value offy in equation (3), we obtain 



{be''y-b'e—''y) a^ {b e'V + b' e—^y) =a{be'y + b'e—'y)a?{be''!'-h'e—''!') 



an identity. 



Thus all the conditions are satisfied. Our solution, then, of the equations is 



/ — » — r''\ 27r 

 u.= ibe'- +(/e ^ \ .sm-^ (c(-z), 



2ir 2!r 



?;= (6e ^ ' —b'e ^ j .cos^r— (c?— a;) . 



If it should appear more general to affix a constant to ?i, it will add no difficulty 

 to the investigation. 



The value of 1/ may be determined by supposing the origin of co-ordinates to 

 be placed at the bottom of the fluid, so that v=0 when p=0 : this process jjfives 

 b'=b. 



9. Our next step is to find p. 



VOL. XIV. PART II. 4 Q 



