PROFESSOR KELLAND ON THE THEORY OF WAVES. 5I5 



Let udx + vdyhQ denoted by dcp, then it is a well known theorem that 



!L. = _,.,_.. ^Jt _!..(,.,,.). 



By means of this equation a value of p is found, which being treated as in 

 art. 10, will give the values of c and bo, &c. 

 We proceed to the most general case. 



« = 6, + b, {e^y + e-'^y) sin O + b^ (e^'^!' + g-2-2/) sin 2 ^ + &c. 

 v=-b,(e^^~e--y)cos6-b,(e^-V-e-^-y) 00826 + &C. 



'^-;^ = - cib^e"^' + e-'^^'sind + b,(e'-^' + e-^'^i' )sm26+ ...} 

 .-. |=-^i' + ^{6.(e''-"+^~"-' sin + 62 (/'*2'+e-2«2/)sin2^ + ...} 



-2 {b; cos2 d + bl COS 4: 6 + ...} 



+ 22'b,b,{e^'-y + e-^~'''y)oosl^s6 



-2l'b^b,(e'^''y+e~^^'^y)G0B7T7e] + P 



■=-gy + c2br{e'-'^y + e-'-'^y)smr6 

 -^bl-b,^b,.{e'-'^y +e-''^y)8mr6 



-| 2 6, b, {e^^^y 4- e-'^^^V) cos^^ 6 



+l2b,Me^''y+e-^"'y)Gos7T^e+p 



the symbol 2 denoting that all the values of b, b, are to be taken, so that ?• mav 

 = 1, 2 . . . and 5 = 1, 2 . . . 



13. Now, since | =/(^,y) -/(^,^) ; 



the first differentiation gives us - ^ = it(^^ . 



° gdx dx 



the second \ dp^ dfjx.y) _ df{x,.) 



Qdx dx dx 



Hence, since the two are equal, we must have ^'-^^^^^ =^,z being consider- 

 ed a function of x. 



