516 



PROFESSOR KELLAND ON THE THEORY OF WAVES. 



Now 



-\b\-b, 1 br (/" - + e~ "■ "-) sin re 



-i 26, 6,(e'-+''«-'+ «-'•+"•--) cos (»--«) e 



+ g 2 6, 6, (e*^' -■ + e-"^ « ^) cos (r + «) 



O=-^^ + (c-6„)«{6,(e''-- + e-"'^)co8 + 2 6,(e2"^ + e-2'')cos2 + ...} 

 -2a{6;sin2 + 26|8in4 6 + ...} 

 + a 2' T^b, b, (/ + '"-" + e-^ + »" --) sin »^ 

 — a 2V + « 6r 6s («'^'~ " " + ^ ^~'' " *) siar + s 6 



^{c-b„)a{b,{e'' -e-"' sin + 2 6. («""*-- e"^'^) sin2 + ...} 



dz 



Tx 



4 a.- — 4 « ;> 



_|62(/--_e-^«--)+26^(e.''"---e-""-") + ...]a 



dx 



- a2' ^ + « 6,6,(/ + "-'-e-'^"'-')oos/--*0 

 + a^ r—s brbg{e —e )cos r + sO 



dx 



dz 

 dx 



or, if we adopt the latter notation, where 2 includes values for /• and .*, which 

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



rt d z . »\-cii/?*«- — r oi z\ a 



0=—g — +(c — 6„) a 2 r 6, (e H- e )cos»-C7 



dx 



i- ^ t i / r + saz , — r + s«;\ • /\ 



+ ^a.l,r — s brb,{e +e )sinr — so 



-a-S.r + s brb, (/"'" + e~^~"") sin r + s e 



, L \ xz-z / T K z — retZs- /\ ^ Z 



+ (c — b„] alb.r (e —e )sai.rv.-— 



dx 



- ? 2 7T7 6, 6. (e"^- -" - e-'^" •") 008 7^76 . il 

 2 dx 



dz 





dx 



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

 l)roblem, as fai- as the assumed form of the velocity expresses the actual state of 

 motion. It wiU readily appear that z can contain only sims of multiples of 6 ; 

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



