534 PROFESSOR KELLAND ON THE THEORY OF WAVES. 



Now, when ^=0 and ax=-^, «=o, and t.=0, 

 Hfl being the value of H when t=0. 



Let ^ = ^' + (e'''~e~'''')(D + cosaxFi + iimax-^i) 



- C^" ' - «~ " ^; (a sin a a: F /- a cos a ar 4/ if) 

 X if- '' + e~ "■ -) (sin « a:// - cos a a; ^ + C) 



hence we obtain, by putting for ~ its value {v) ^, 



(^"^ — e~"0 (cosaa:/; + sina;c0('+H) 

 X {a(f''- + e— "-~) (D + cosaa;Fir+sinaz-4/;)-l } 



= (^ -^ ) 1-^ + cosaa;F7 + slna.r^}/'M 

 -a(^2«.-_^-2««-) (sinaa:F^-cos«ar4.^) 

 X (sin a X ft— CO?, a.x(p f + C). 



Equate separately to zero the coefficients of e"^-^-"^, and of e^'^' -e-'^"', 

 and there results 



(cos ««//+ ain a.xcf)t+'H.) (D + cos a ;i- F ^ + sin a a; -vl/ ^) 

 = — (sinaa;F^— cosaar-v)//) (sinaar/^— cosaa; ^+ C) 



and cosaa;/^ + sinaa;0;+H=--- cos a a; F'/ -sin a.r-v)/' iT. 



From the former equation, we obtain 



D (cosaar//' + smaa;0?)+cos2aa;/^F^ + sin2aa:0/-v|/; 

 + sin « a; cos a a: (/; .v)/ i( + F ^ ?) + DH + H cos a a; F if 

 + Hsinaar'4/^ = — C (sin aa; F ?— cos aa; -^ t) 

 — sin2 a a; F tft— cos^ a a; ^ %}/ <f + sin a a; cos a a: (F ^ ^+// -v). () -. 



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



D/^+HF^=C.v|.^ (1), 



li(pt+B.-^t= -CYt (2), 



ft-^t+'Ftcpt=Yt(pt-^ft-^t an identity; 

 '2J)^+ft¥t+<l)t-^t=-Ytft-cpt-^t 

 or DB.+/tYt+(pt^t= (3) 



ft¥t-(pt^t=Ft/t-cpt^t an identity. 

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



■?7 = -H (*) 



