540 PROFESSOR KELLAND ON THE THEORY OF WAVES. 



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

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

 obtain 



= -6acos0 (e'^Vj^ir'y) { (J g" ." + e" " •") (1 + sin 0) - c 1 



+ a 6-^ (e- ■" - e- - ■")' (1 + sin 0) cos 6 

 = _„/!,f (l+sin0) cosd A + cab cos 6 (e'^ + e-^y) 



Y-^=~.9+ {-6.°"-.-"^|8inea(^-.) + « 6 cos (e" 3' + « — ■") (^) } 



= -g-absm6 {b 1 +sin {e^-V -e-'^°V)-c (e'^'-' -e" 'V) } 



-ab'cos'eie^-'^'-e-^'y) 

 = -ff-ccb' sine U'^''y-e-^''V)-atP (e^'y-e-^'V) 



+ abc sine (€""-6-'^) 



'LL = -ab^ {4:COse + 2sm-2e) dz + cabcose (e'y + e-'y) dx 

 ? 



-gdy-ah' (l+sinS) {^'■y-e-'^''y)dy + abc{fV-e-'-y) sin r/ ;/ 



= c 4 (/.(«"•" + «-«!') &me-gdy 

 ^ = bc(e''y + e-''y)sme-gy + V 



9 

 where P is a function of z. 



d d) X z dtbxz dz /^ , . , , . « 



Applying the formula ^^ + ^^ Tx wl"ch we proved, art. 9, we 



obtain 



= 6 r a (e"V <-""'*') cos + (-^ + 6ca sin (e"''- <?-«^))^ 



and from the value 

 we obtain 



,^A + A(e»^_e-»--) (l + sin0), 

 a C 



li = -(e"-'-e-"-')cose +^(e»-'+e-«--) (l+sin0) 

 dx c c 



— (e — e ; cos D 

 c 



1_* («"--+ (^- -')(! + sin (9) 

 e 



By substituting this value in the above equation, it becomes 



*- (e« •"-«-"-') cos 



6cB(e°-+e— "-) COS0+- {-g + be a. im 6 e" -e—")=(i 



