PROFESSOR KELLAND ON THE THEORY OF WAVES. 523 



(c - 6„)= 2 r a, (e'' "'' + «-'•«*) 



= -^ 2 r «r (e — e ) • 



a 



From this equation, we obtain 



2ra,(e'''"'' + e~ '' "'') ■*- a' 2 r' s i Ur a, a, x 



' I _ (e/« '' + e— /« '^ ) - (e(/— 2 O a A ^ e-(/— 2 <•) « Jj ^ //— 2 s) « A ^ ^ -( /- 2 s) « /, 

 + e(/-20«A + e-(/-20«''| ~| . 



22. Since v must of necessity be zero at the points where « is a maximum 

 and a minimum, we get for the values of 6 at such points, fu'st from the value of 



^ being at such points, 



0=a, {e'"-e-") cos + 2«, (e'^'--e-'^'") cos 2 + ... ■ 

 and from the circumstance that ^=0, 



0=b,{e''~'-e-''-')cose + b,(e^'"~-e-^''-')cos26 + ... 

 But by art. 14, 



b, = a (c — b^ a, 

 b., = a, (c — b^ 2^2 



therefore, the second equation becomes 



Q = a, (e'-'-e-'-') cos 6 + 2 a., {/''- -e-'^"') cos 2 + , . . 



which is identical with the first. This is strongly confirmatory of the correctness 

 of our operations. 



If (as is probably always the case) the wave be symmetrical on both sides, 



we must have this zero occurring at points distant from each other by -„- ; hence 



TT 3 TT 



= .^, -^, and a-i, a,, &c. =0; 62, 64, &c. =0. 



The value of u at such points is 



tt = b^+b^e 





sin d + 63 e 



-—y 



sin 



3^- 



TT 



'' 2 



Stt 

 ' 2 



2a- 6<r 10=r 



= b„ + b,e " -b,e ^ +b,e '■ - &c. 

 = b,-b,e '■ +b,e ^ " - &c. 



