540 PROFESSOR KELLAND ON THE THEORY OF WAVES. 



We proceed to the determination of the motion. All that remains for us to 

 do, is to substitute the values of u, v, and z, in the equations of art. 8. We 

 obtain 



^- ^ = -6 (e^^' + e-'^) («cosa)(^-o'\ + « 6^ (^" •'' - e" " -^ cos ^ (1 + sin ^) 

 ^ dx \dt / / V / 



= - 6 a cos (e"J' + e-'^V) { (6 «" ." + g- « 2/) (1 + sin 0) - c } 



+ a62(e"2'-e-''2')2 (1 + sin^) cos ^ 

 = -a6«(l + sin^) cosO.4 + ca6cos0 (e^^' + g— "y) 



^^^= ~,g+ i-be''^~e-^v\smda(^-A + a b cos 6 (e^V + 6-"^) (^\ | 



= -ff-absinS {bl + smS (e^'^V -e-^^^')-c (e^V -e-'V) } 

 -ab' cos' d{e^''^-e-^''V) 



+ a.bc sine {fy-e-'^y) 

 .: ^ = -a.U' {^ cose + 2 smiQ) dx^-c ah cos e if y ^ e-'^y) dx 



-gdrj-ab' (1 + sin^) {e^'^V -e-^'^V) dy + abc (fy -e-'^y) sin 6 rf^ 

 = cbd.{e°'y-\-e~''y)sme-gdy 



i- = bc{e''y + e-''y)sme-gy + ¥ 



where P is a function of z. 



, . TO 1 d(bxz dd)xz dz ^ ^ . . 



Applymg the formula — ^ — + — ^^ ' d~ which we proved, art. 9, we 



obtain 



= bca(e''' + e-^') cos + (-^ + 6ca sin ^ (e«^- «-« ^)) — 



and from the value 



we obtain 



, = A + A(e«^_e-«^) (l + sin^), 

 a c 



dz b f i^z — tt.Z\ /J ,6/aZ, — a.Z\ r-i , „• /J\ 



— — — [e — e ) cos a H — ie + e ) (1 4- sin a) 

 dx c c 



" r a. Z — a Z\ n 



— ie —e ) cos o 

 c 



1_* (e«^+^-"^)(l + sina) 

 c 



By substituting this value in the above equation, it becomes 



b , u. z — «^\ /> 

 - (e —e ) cos V 



c 



bca(e'"' + e—'") cos0+ r (-^ + 6c« sin ^ e"^-e-"^) = 



c 



