PROFESSOR KELLAND ON THE THEORY OF WAVES. 535 



ft = -¥'t (5) 



<f)t = --^'t (6). 



34. Let us next solve these six equations. To find C, we must combine 



(1), (2), and (3). 



From (1) and (2), 



C((t>f'^t+ftFt) = B((ptFt--^i//) (a) 



and from (3), 



-CT)n = G(feFt + <l)i-^i) 



-CJ) = (<ptF(--\,tf() (6) 

 To find H, we eliminate C by (1) and (2), and obtain 



H[(Fty+(-^/y}+j){ftFi+cp/^f)=o (c) 



Putting for H its value from (3), we get 



Also, by (5) and (6), 



fiF e+(pe-^e = -(FiF' i + -^i -^'t) (e) 



= -1^7^^^ 



dt 



But by (3), 



f/F t+(f> i-^ t =-Dn 



BH = 0,f/Ff+cpt-i,(=0. 



35. We may satisfy the equation DH=0 in two ways: 1. by making D=0, 

 in which case, by virtue of equation {d), both F^ and -\'t equal zero, or z is itself 

 constant. This case coiTesponds to equilibrium, and we have nothing to do with 

 it. 2. By making H=0 ; which hypothesis requires that D shoidd be a constant, 

 independent of the time, by equation (4). To prevent error, in consequence of a 

 quantity equal to zero appearing in our equations, it will be desirable to write 

 them down again, omitting H. We shall also omit t for the sake of brevity, and 

 write/ instead of ft, <p instead of4>t, &c. 



The equations are 



D/=C^^ (1) 



D^ = -OF (2) 



/F + 04 = O (3) 



/= -F' (5) - 



(p = -+' (6) 



The equation (3), combined with (4) and (5), gives 



