PROFESSOR KELLAND ON THE THEORY OF WAVES. 515 



Let udx + vdy be denoted by dcf>, then it is a well known theorem that 



By means of this equation a vahie of j9 is found, which being treated as in 

 art. 10, wUl give the values of c and b„, &c. 

 We proceed to the most general case. 



« = 6, + 6, (e- .'' + e- « !') sin + 62 (e- « .'' + e-2 « y) sin 2 e + &c. 



u=_6,(e«y-e-«2')cose-62(e2«.''-e— 2".")cos2 + &c. 



(/, = 6„_l|6,."^+.-'.".cos0+|.(.2«-'' +«-2»;')cos2 + ...} 



1| = - c { 6,/-"+e-"-"sin d+b, (/"'■' + e--'' ■" ) sin 2 + ...} 

 -I rbl + 2b,[b,e'y+ e-'y sin O + b, (e^ «V + ^-^ « !/) sin20 + ...} 



+b-;?^y+7-^y+b\{e*'"j+e-^'y)+ ... 



-•2{b^co&2 6 + hl cos 40+ ...} 



+21: b^b^ie'^^'^y +e-^^"'y) cosies e 



- 2 2' 6, 6, (/-" 2' + e-^ « ;') cos F+7 J + P 



= -^^ + c 2 6, (£'■«:'+ e-"^) sin »■ 

 -^bl-b^^Mfy +e-"^'J)smre 



-^lb,.b,(e'-+"'y+e—'*"'y)oosr-se 



+ ^lb,b,{e'^°'y+e~^"y)cos7T76 + P 



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

 = 1, 2 . . . and s = 1, 2 . . . 



13. Now, smce ^=f{^,^) -/(x,^); 



1 dp_ df{x,y) 



the first diiferentiation gives us 

 the second 



q dx dx 



ldp_ df{x,y) df{x,z) 

 qdx dx dx 



d f(x 3) 



Hence, since the two are equal, we must have ' \ =0,2; being consider- 

 ed a function of x. 



