PROFESSOR KELLAND ON THE THEORY OF WAVES. 5^3 



or ^ c^ (e" * + e-« '0 1 1 - {^ a \ (e« '' - e-'^ ^J \ ^g (e« '' - e'"- ^) 



27r „ ah_^—«h 



C2 — ^ . 



e"" + e- 



!-(-.)(.' 



277, 27r , 

 = ^ o^ nearly 



hence c^=gh. 



11. Our result has been obtained from equation (9) by putting 6=0. This 

 mode of proceeding, of course, gives only one result ; we shall, therefore, pursue 

 another process in order to obtain a second. 



Since z=h + a (e«* - e" "*) sin 6 



— h + m sin 6 SUppOSe, 



let us substitute this value of z in the equation (9), and expand the exponential 

 functions. 



It will appear readily that 



277 



e«^ + g— «« = e"'' + e—"^ 



-^»«sinO(e«^-^-«'^) 



e"-" - e— «^ = e«^ - g— «* + ^m sin (e«^ + e" "'') 



if we omit powers of m sin 6 greater than the first. 



But equation (9), by means of these values, becomes 



+ ^b' sin2 (e«^ + e-« *) - ^ 6 c sin (e" '' + ^-« *)2 

 ^ '^' 6 c sin2 a a (e" ^ + e-« *) (e« '* - g-" *)2 = 



^ (e«* -e-«'') +^^a sin^ (e^"'' - e-2«*') 



_?^ 6 c sin (e"^ - e-«'')2 - ^ be a sin^ ^ (e*^ + e-«^) (e«^ - e""*)* 

 A A^ 



27r r ,.A _„* 2'7r 



A 



,2/ ^«A _ g-«A + f^ « sin e (e^ «'' - e-2 «^) I 

 X I e^-l' - e-2«A +i^a sin ^ (e^"'^ + e-2«*) (e«* - e-«^) I 



This equation will furnish the two results mentioned above. The first, de- 

 rived from the parts which do not contain B, is 



A ' A 



VOL. XIV. PART II. 4 R 



