536 PROFESSOR KELLAND ON THE THEORY OF WAVES. 



F' + -4'''= a const, 



= E'' suppose : 



but F + 4' = ^ {f + (p--) by (1 ) and (2) ; 



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



If F=Esin7, ■v|/ will =Ecos7; 



36. Again, since -^^VE'—F' 



-FF 



G" E'^ 

 by substituting which in the equation F" + -^"= -^ we get 



or 



F 1 



and F=Esin(i/Cf/?y 



Similar values may be obtained for the other quantities /, (p • • • ^ ^'^^ thus the 

 number of arbitrary functions wUl 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-^'{/eoBact+ffsmact+k) 



■^ t—er^'(p cos a,ct+gsmac(+r) 

 ft = m e-"" (/cos . +^ sin . +k) + ac e-"" (/'sin . —// cos .) 



— e-^'(m/—acy. cos. + mg + aef. sin . + mk) 

 and (/) / = ;«e-°"(;)cos .+ysin. + »-) + ace-""(;^sin. -ycos.) 



= 0'^' {mp — acq COS. + mq + a.cpsin. + m r\ : 



by writing cos . for cos act and so on for shortness. 



