516 PROFESSOR KELLAND ON THE THEORY OF WAVES. 



Now f(x,z)=-gz + c2br{e"'~ + e-'-''')%mre 



1 



- i 2 6, 6, (e''+* « " + e- »•+* « cos (r- *) 6 

 -2a{6;sin2a + 26*sin4^ + ...} 



+ al r — sb,.bs{e +e )sinr — sd 



— a2r + sbrbs{e +e ) svar + s u 



-t {c-b,)a{b,{e'^' - e"^' sin ^ + 2 b, {e'^"' - ^-^"^)sin2 ^ + ...} — 



dx 



-{bfie^"' -e-'^^') + 2b\{e^''' -e-'''^') + ...}a~ 



dx 



T/ . i 1 f r + Stt.z — r + Stt.z\ /» dz 



al r + s brbs(e —e )cosr — s0 — 



dx 

 d i 



r — s a. : — r — s a z\ /» a « 



+ alr — sbrbg(e —e )co8r + sd 



dx 



or, if we adopt the latter notation, where 2 includes values for r and s, which 

 may be identical, which 2' does not ; we get 



1 



+ -oi.lr — s brbg{e +e )sinr — sd 



-ta2r + s b,b,(e''-'''' + e~^-''') sinr + s d 



, / iN V i / ^«-? — r«Z\ •_ /J dz 

 + (c — bo) a z bf r (e — e )8inro.-—- 



dx 



2 dx 



dz 



. "' -^ J 1 /■ T — ^s a z — r — s tt.z\ , a ""^ 



+ --lr — sbrbs(e — e ) cos r + s v . -— . 



2 dx 



14. From this equation we shall be able to obtain a complete solution of the 

 problem, as far as the assumed form of the velocity expresses the actual state of 

 motion. It will readily appear that z can contain only sines of multiples of 6 ; 

 for, if it could contain cosines, v would also contain sines, which it is supposed 



