534 PROFESSOR KEIXAND ON THE THEORY OF WAVES. 



Now, when /=0 and «*=— ^. «=0, and«=0, 

 H„ being the value of H when /=0. 



Let * = /« + («"* — e "-) (D + cosa^F/ + siiiaa--4//') 



— (e" " — e~ "') (asm ax Y i— a cos ax -^i) 

 X (e"' +e^"') (sma»-/7 — cosax (p t + C) 



(I z 

 hence we obtain, by putting for -^ its vahie (?')„^_, 



((?" • — e~ " ■^) (cos a a:// + sin a .!• (^ / + H) 

 X {«(«"•+«"""") (D + cosa^F/ + sinaa,-N)/<)-l } 



= (p"^-'e~''-) (^ + cosaa;F7 + sina.r^'A 

 X (sin a xft—co» axcp t+C). 



Equate separately to zero the coeificients of e"-e~~''\ and of e'-''--e^^"% 

 and there results 



(cos a «/< + ain « a; ^ < + H) (D + cos a .f F / + sin a X "4/ if) 



= —(sin a* F / — cosaX'v)//) (sin ax ft— cos ax <f) i+G) 

 and cosaa://+sina«0 / + H= — -T cosa.r F'/— sinax^'Z- 



From the former equation, we obtain 



D (cos a*// + sin ax(p l) + cos^ax/t F <+ sin^ ax(p t-^t 

 + saia,x (MSax{ft-^t + V t(pt) + DH + H cos a a; F / 

 + H sin a a; ■4' < = — C (sin a a; F if— cos a a; %]/ ^) 

 - sin^ axV tft— cos- az(p l-l'( + sin a ar cos a a; (F < / + /7 •+ : 



in which, if we equate the coefficients of sines and cosines of a.r. 2 ax, &c.. we get 



D/<+HF/=C-+? (1). 



D<pt + H'si^t= -CF< (2), 



/(■^/+Ft<pt-F((pt+/(-^t an identity; 

 •2DH+/IF t+(pf^/= -Y tft~(pi-^t 



or DH+/'F(r + ^/4/ = (3) 



ftY t-(f>t-^t = Y t/t-(p i-^( an identity. 

 From the second equation, we obtain, in like manner, 



S=-" « 



