512 PROFESSOR KELLAND ON THE THEORY OF WAVES. 



(l-^.ae^-fe-'''sme\ b cos 6 (e"' - e-'') = 

 ^ cos 6 (ae* ' + /e-"^) {c-6 sin 6 (€"'+ e-"') } : 

 or b (e« -'- e-''')-^b (6« ^- e-"') (ae" "-/^-" ') sin 6 



= 'i^c(ae'''+/e-'')-^bsmd(e'''+e-'')(ae''' + /e-'') 



f)_. 



Hence /=-« and b=^- ac: 



A 



dx 



dz 27r n , . 



= — -^a cose' (e^-' — e— «^) 



A aa; 



^^ o cos (e*^ — e""^ 



which l)eing substitnted in equation (8), reduces it to 



■^'^ 1. I \ u ~:„ a f .a.z _ — tt.z\ \ /\ i t b ' . 



-^ b Ubame-{e-'^ "-''') c\ cos 6 (l--sin d (e'^' + e-'')\ 



= - cos^ {e'~- e-'^') I (f-^ be sin 6(6'''- e-'^') + ^ 6^ (e2. = _ ^-2«^^ | 



that is 

 27r6 



|46csin0-c2(e«^+ «-«-')- 4 i^ sin ^ ^ {e"^ + e-'=) + be sin 6 (e" + e-")' ] 



jLige'^'-e-'^' -^ be sin (e«-'- e-'^J + X^' ^^"'~ ^""^^ ^^^"'"~ ^~'^'"^ I 



or, therefore, 



-S^ besmB + ^-^c" {e''+ e-'') + ^ 6^ sin^ 6 (e'' = + e—') -^^besmO (e"+ e-'-J 

 A A A A 



_ _j7 (e«^ _ e-«^ _ ^ 6 c sin 6 (e''- e-'^f + x" ^' (^"^ " ^~"0 (^ - ^""^ ") (9) 



Let 0=0, then z=h, and the equation gives 



A A ^ ' 



