or 



PROFESSOR KELLAND ON THE THEORY OF WAVES. 5^3 



25^(e./'+e-«A) |i_ (?l.ay {e"!' - e-"'')'] -vie"'' -e-'l') 



C2 = ^ . 



A f,«.h , p—uh ' /27r \^ 



e"'" + e 



1_(^ ay (,./'_, -.A). 



= 3 ^ nearly 



hence ri=gh. 



11. Our result has been obtained from equation (9) by putting 0=o. 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{f^ — e— » *) sin Q 



=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 



e" '- _ e— « = - e" A _ e— "* + ^ ,„ sin («« '' + g— « '') 



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



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



+ ^tP sin2 (e«* + c-«*) _H^ 6 c sin 6 (e"* + e—''')' 

 A A 



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



J^bc&md (e«*-e-''')2_^ be a sin^ 6 {e''' +«-"*) («"* -e—*)? 

 A A" 



+ l!r6J ,«A_,-«A + ^„sin0(.2"'' -.-2»'0l 



+ 1^ a sin (e^"'' + e-2«'') {e-'' -«-»'') I 



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

 rived from the parts which do not contain 6, is 



VOL. XIV. PART II. 4 R 



