PROFESSOR KELLAND ON THE THEORY OF WAVES. 517 



not to do. Let us then assume 



z = h + a, (e« - sin 0) + Os e^ " - sin 2 + &c. 

 +/ e— " ^ sin +/2 «-2 «■ ' sin 2 + &c. 



.-. ^=a»r^[(a,e«-+/e-«')cos0f2(aoe^"-+/,e~^''-)<'os20 + &c.} 



+ a^^{o,(e«^-/e-«Osin0 + 2(aoe2«.-__;|e-2«^)gin2 + &c.} 



dz 

 or substituting for -^ its value 



( — 6, e"— c cos — fij. e "■* — e "^008 2 — &c.) x 



{!-«.(«,« -/esin0 + 2.o,e^"-/,e-'""^8in2e+...)} 

 = a{(a,e'*^+/e-"Ocos0 + 2(a,/"-'+/,e~"^''')cos2e + ...} 

 .|6„_c + 6,(e" + e-''^)sin0 + 6,(e2«'*-+e-2«^sin2 + ...}. 



By equating coefl&cients, we obtain 



f=—a,,fi=—ai ... 

 aa,{bu — c)=—b,, ... 

 2 a aj (6o — c) =: — 62 J — 

 &c. =: &c. 



.-. aa,{c—b„)=b, 

 2aai (c — b^ — b-i 

 3a«3(c— 6„) = 63 



&c. = &c. 



So that «=A + a, (e" '-«-"") siad + a,{e'^'"'-e-^'"^ sin20 + .. 



d z 



and ^ {1-a (a.e'^+e-"^ sina + 2a2. 6^"^+ e"^"'' sin2e+ . . .)} 



ax 



15. Substituting this value in the equation, we get 



r (c-b„) a. {b.e'^+p-^ cose + 2 b^ie^"" + 6-^"^) COB26 + . .. ] 

 + ^lV^sbr b, (6^"'+ e-'^'') sin^^^e 



{l-a(o,e"^ + e— ^sine + 2a,e^«^ + e-^"^8in2 0+...)} 

 VOL. XIT. PART II. 4 S 



