520 PROFESSOR KELLAND ON THE THEORY OF WAVES. 



+ r(/-^ + ^«" + e-^-^ + ^"Ocosr + *-^0 



s {e +e )oosr + s + t6 



+ s (e'-'-* '"' + 6-^^^^^"'') cos r + s-td J 



Or since 



2rcosy-*-^0/+* + '«^ 



&c. &c. 

 this term may be written 



"26 



-' Ib^b.bt j -2y(e''+« + ««^+ e-^ + * + ««-^)cos« + ;-r0 



+ 2y(/+*-*^^ + e-^+*-*"0cosr+;-*^ 





which may be written 



a^a 



= -^2rbrb,b,{ - (e/«^+e-A^cos/-2r - (g/- ^'"/"^ + e-/-2'-/«^)cos/d 



+ (ef-^^/'" + e-f-^*^<'')coa(/-2s). ^ + (e/-2V«^ + e-/-2*/«^) cos (/-20 6j (3) 



if r+* + # be denoted by/ 



19. The remainder of the lower side of our equation is 



•^' 2 J, (e*- « ^ - e-*- « ^) cos /• . 



By collecting these several terms our equation is reduced to the following 

 very simple form : 



(c- bo) a2 r brie""" ' + €-''''') cos rO 



r + saz . — r + s«g< 



+ az{e +e JTb^b^ sm r—so 



— az[e +e ) y o^ Oj sin r + * a 



a . 



+ ^2(r-*)6,6,(/+*"%e-'-+''"0sm/-*a 



a 



-'^lr + sb,b,(e'-"'' + e-'—'''')sm^T76 



