KUNGL. SV. VET. AKADEMIENS HANDLINGAR. BAND 47. NIO 4. 



L3 



provided vve put: 



i o 



■/.') +2 - .'' ""'. W ' . -', 



sinh fcÄ . J v 



k C 



kc \ I Le . 



COS [vat— sin o/ | ( l)''cos ;'«/ sin o/ 



'"(/) 



— Z • i ,i . ■■■''■{ SIM ''"'' sinat] — ( — 1'sm w(+ sm at\ 



—* smh kh . I, \ a / L \ <f I \ a ' 



c— i 



Eere 



/,. = v' 1 o- o (coth kh + coth M') — gk {q — (/) 



and, since o — e is very small, we ha ve: 



J v = v 2 a 2 a — gk{g — o) . 



[ntroducing these expressions in the system of differential equations, an easy cal- 

 culus will give us: 



A (v, <M - - 2 t 'Zul-i, ■'' (v) [ cos H-T sin "') + ( - ' '* t- " s I '"" + T sin "')] 



*«-•«- ^^;;;^^(^h('--*; si ''''')-<--'>-H-^;--)] 



Taking the first of these formulse we write it: 



D, (<p, W) = ^ — r-PW • 2 cos I- sinff^l Y 4>'-o' 2 .«/ 2 v — I cos 2*>ffJ — 

 17 « sinh kh \a ' Jmt \ a ] 



-5^ H -»-»(T' i »" , )2<*' + 1 ) , « ,J '^ (t) •'»«'» + «'«• 



Now, taking the second derivative with respect to ti me of the identity: 



x — sin (jn = 2 cos fc.r . ^ Ja.,. I I cos 2 vät + 2 sin kx r J-iv+\ I I sin (2 >' + 1) at, 



o o 



and putting afterwards 



x = - sin <j<, 



we evidently obtain: 



D.(rp, 4A = — k c 2 cos 2 at. . ^ g T 7 = fcc 2 cos 2 a< . O. 



smh &/& 



In a similar way, the second equation will be seen to be satisfied because of the 

 identity: 



(\ 0D / °° / 7 \ 



x sin at\ = 2 sin kx 2«^2r ( — ) cos 2vat — 2 cos kx^J^v+i — I sin (2 ■»' + l)a< 



