PROFESSOR KELLAND ON THE THEORY OF WAVES. 517 



not to do. Let us then assume 



+/e-«^sin^+/2e-2«^sm2 + &c. 



+ a^A a,(e''^-f,e-"') sin e + 2 (a,e^ ''^ -f.e-^'^') sm2 + to.} 



€v t 



dz 



or substituting for -r- its value 



( — 0, e —e coso — b^.e —e cois2 a — &c.) x 



{!-«.(«,/ -/e sin a + 2.a,e^"^-/,e-^"%in2^ + ...)} 



.{6^_c + 6^(e"Ve~"0sina + 62(e2«^+e-2«^gin2 + .,.}. 

 By equating coeflBcients, we obtain 



a,a^(ba — c)=—b,, ... 

 2aa2(6o — c) = — 62, ... 

 &c. = &c. 



.-. aa,(c—bo)=b, 

 2a,ai{c~ &(,) = 62 

 3 a as (c— 60) = 63 



&c. = &c. 



So that ^=-^ + a, (e"^-e-"^)sma + a2(e2«^-e-2«^)sin2 + . 



rf« 



and -T- {l-« («.e«-'+e-"^ sin^ + 2a2. e^"^+«-^«^sin2 + ...)} 



ax 



15. Substituting this value in the equation, we get 



{1 - a (a, e"^T?-«^ sin ^ + 2 a^ e^^^+7-^^ sin 2 a + . . .) } 

 VOL. XIT. PART II. 4 S 



