536 PROFESSOR KELLAND ON THE THEORY OF WAVES. 



F^ + ^vj/'^ a const, 



= E'^ suppose : 



but F^ + ^^=^\/^ + 0^) by (1) and (2); 



or F" + ^"=^ by (5) and (6). 



If F=Esin7, -v^ will =Ecos7; 



\dj) ~D 



36. Again, since ■x}/=\/E'— F' 



Ve'-f^ 



C'E^ 

 by substituting which in the equation F'' + -v|/"= — - we get 



j,,^ F' F" C E' 



or 



E'-F^ W 

 F' _ C , 



and F = Esin 



(|/c.,) 



Similar values may be obtained for the other quantities /, (p . . . , and thus the 

 number of arbitrary functions will be reduced to one, viz. C. This is the general 

 solution of the problem. - 



For the present, however, we prefer the examination of the following parti- 

 cular case : 



37. Let F t=e-^'(f coaact+ff sin uct+k) 



'^t=e~'^'(pGOSac t + qsinac i + r) 



/t=me~'^' (fcos . +^siD.. +k) + uce~^^(/sm.~ff COS.) 



= e~"" (mf— cLcg . cos .-^-mg + acf . sin . + mk) 

 and <p i — me-'"'^'(pGOS . + g sin . + r) + a c e—""' (p sin . —qcos.) 



= e"~"" { mp — acq cos :+mq + a, cp sin . + m r] 



by writing cos . for cos a c ^ and so on for shortness. 



