452 Mr. stokes, ON THE THEORY OF OSCILLATORY WAVES. 



The general value of (p^ will be derived from (13) by merely writing — h for h. But in order 

 that (32) may be satisfied, the value of <j)^ must reduce itself to a single term of the same form a? 

 the second side of (35). We may take then for the value of (p^ 



^, = ^, (e""*+'" + 6-'""^''') sin tnx (S6). 



Putting for shortness 



6 



+ e =o, 6 — 6 = Lf, 



and taking S^, D^ to denote the quantities derived from S, D by writing A for h, we have from (32) 



DA + DJ^= (37), 



and from (33) 



p(gD-mc^S)A +p, {gD, + mc'S;)A^= (38) 



Eliminating A and A^ from (37) and (38), we have 



g {p-p)DD 



mpSD, + p_SD ^^^^• 



The equation to the common surface of the liquids will be obtained from {Si). Since the mean 

 value of y is zero, we have in the first place 



C.= C (40). 



We have then, for the value of y, 



y = a cos mx (^Oi 



w 



here 



mcpAS-pAS DD pA S - pAS , , 



a = — ' ^' ' ' — ^ = '' ' ' ' — '- (42). 



g p-p, c pSD^+pSD ^ ' 



Substituting in (35) and (36) the values of A and A^ derived from (37) and (42), we have 



ac 

 D 



^ = - ^(£""*-;" + e""i*-"i)sin»M.r (43), 



0, = ^ (e""* ^•*' +£-"'<*.*-i") sin mx (44). 



Equations (39), (40), (41), (43) and (44) contain the solution of the problem. It is evident that 

 C remains arbitrary. The values of p and p^ may be easily found if required. 



If we differentiate the logarithm of c^ with respect to m, and multiply the result by the product 

 of the denominators, which are necessarily positive, we shall find a quantity of the form Pp + Pjj^, 

 whei-e P and P^ do not contain p or p. It may be proved in nearly the same manner as in Art. 4, 

 that each of the quantities P, P^ is necessarily negative. Consequently c will decrease as m increases, 

 or will increase with X. It follows from this that the value of (p cannot contain more than two 

 terms, one of the form (3.5), and the other derived from (35) by replacing sin mx by cos mx, and 

 changing the constant A : but the latter term may be got rid of by altering the origin of x. 



The simplest case to consider is that in which both h and h' are regarded as infinite compared 

 with X. In this case we have 



d) = — ace "'" sin mx, m, = ace"'" sm mx, c = - — -' — , y = a cos mx, 



p + pm 



the latter being the equation to the surface. 



