PROFESSOR KELLAND ON THE THEORY OF WAVES. 129 



By altering -^M into 0' (m) to make the notation similar to that usually 



cos m k 



adopted, we get 



" ~ ) cosm (*-*) rfi 



-)cosm(x-k}dm . . (6) 



This value of (/>, it must be observed, has been deduced from the general 

 form by the aid of equation (3) which belongs to the surface, and as that 

 equation is only an approximation, so is this value of < itself only approximate. 

 The object of the following process is the determination of the functions $' and 4-' 

 by means of a knowledge of the initial state of the fluid. 



67. Let us suppose that the motion has been produced by the sudden loosing 

 of a disk which kept a small portion of the fluid at a higher level than the rest. 



The integral of the equation for the pressure is p= -gy--~&c. + C, 



therefore at the surface = -g z - - + C, 



or 



the density being 1, and if + v 2 very small. 



Also <, is the impulse on the surface, therefore when t=0, $,=0, -'= 



, 



it=:a,y=b, when b= '-- is the original form of the surface, 



-*)<* "* ' ' (7) 



and <f)' (m)=Q. 



Thus the functions </>', </>' y, -47,, are all determined. It remains that we ex~ 

 press the function 4' in terms of the given function /or its equivalent. 



Denote ak by a',f(a) by/ (a') ; x k by of ; 



then /(<*)=* ^ (e' b + e~ m 6+2 A ) # (m) cosm Of dm 



. . (8) 



and our object is to obtain in terms of /without the intervention of -4/; or to 

 eliminate 4*' between these two equations. 



68. We shall make use of the following theorem in order to effect this pur- 

 pose. 



THEOREM (See CAUCHY'S Memoir, Note xi.) 

 /> 



If 2 / cos a' m <$> (m) d m =/ (a'), then will 



\J 



VOL. XV. PART I. Mm 



