PROFESSOR KELLAND ON THE THEORY OF WAVES. 50,9 



Hence the equation ~ . (g) = ^ • (^) gives 



ac.cos6/>-2asin6 cos 6/>/'y + sin ^ cos (Fy f" y + V y f'l/) 

 = a^ c Fy cos - 2 a^ Fyfy sin ^ cos + a F'y F.y sin 2^. 



Equating coefficients, we obtain 



f'y^a¥y (2) 



-«/>/'y + ^-•^•^■''■?'|-^-'^^ + «« Yyfy-Y'y Yy «=0 



or -- F'.y F".y + J (- Fy F"> + 1 Y'y ¥"y\ =0 



or 



a 



F">--FyF"y=0 



or 



¥y ¥' 



"y 



= ¥'yWy (3) 



but 



'•' 





/>=«Fy 



and 







«F>=«2/y 





•'• 





f"y=<^'fy 



or 



• 





''fy-^Kf. 



dy' --■"' 



The complete solution of this equation is 



fy^he'y^b'e-'-y 

 F> = «(6e*2' + 6V-"."); Fy = 6e"2'-6'6^-"i' (2) 



If we substitute this value ofy ^/ in equation (3), we obtain 



{he'^y — Ve—^y) a? {b e'^ + b' e" «^) =a{be'y + be—^V) a? {b e^y — Ue—"'-') 



an identity. 



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



— y ^\ 27r 



be'^ -\-b'e ^ J . sin -^ (c^— ar) , 



v=z (be ^ —b'e ^ \ .cos,-^{ct—x) . 



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

 to the investigation. 



The value of h' may be determined by supposing the origin of co-ordniates to 

 be placed at the bottom of the fluid, so that ^=0 when ?/=0 : this process gives 

 b'=K 



9. Our next step is to find p. 



VOL. XIV. PART II. 4 Q 



