PROFESSOR KELLAND ON THE THEORY OF WAVES. 511 



dcbfxz) dz dcbixz) ^ ,_, 

 hence ~ V • i~ + , =^ (< > 



dz dx dx 



1 0. To prevent confusion, however, we will at first write do-wn all the terms 

 of equation (6), and afterwards strike out those which occur on both sides, so as 

 to obtain the form of equation (7). 



J ^ ' g dx 



and by (5) 



1 ^ i^_^l,^QQ^Q(.^y ^e-'^y -e^'-e-'^') 

 Q dx dx 



^ ' dx ^ ' dx 



hence -a.bcao^Qie'^y ^e-^y) +262asin2 



- aficsin 0(e»^- «— ^ ^ + 6'« (e2«^ _ e-2«^) ^ 



w 3; dx 



that is, as we should have deduced at once from equation (7), 



+ -^-^ — sin 2 f = 

 A 





or -^.^n...-^ 



A 



/ ?5, —if- 



2!r 25r ^ 45r 4? 



2'7r. . . / 



■ —- \ q — ^— c sm f I ^ — ^ 



[e^'-e >-'\^-^i^ie^' -e ^'U (8). 



(25r _J^_^ \ 



a.e'^' +fe ^) smd 



2* 2?r 



2 5r_ 2_^^ 2^^ 2 5r 



+ (a ^~' +/e~~' ) ^cos e L- b{e^\ e~~' ) sin 6 \ : 



But from the cu-cumstance that ^=^ when y = z, this gives. 



