PROFESSOR KELLAND ON THE THEORY OF WAVES. 



50c 



dv 



du 



queiitly j^ , which is equal to - ^ ,* is also independent of y. 



V will in consequence assume the form v—yf{x, t) + cp{x, t) ; but when y—Q, 

 V is always =0, hence v=i/f(x,t) ; and, by reference to Art. 2, it appears that 



1/ da, 

 a. dt 



But 



' dt ' 



hence 



Cor. 



dy _ y da 



dt a, dt ' 



dz z da, 



dt a dt 



u dz 

 «) = -. — . 

 z dt 



1 dz _ 1 da 



z dt a dt 



4. Now z has been assumed to be equal to h + a sin Q. 



dz ^'Kca n 2'Ka , 



-;-= — r — . COS a ^ — .cost 



dt X X 



V 27!' c a n 27ra?/ 



V =- ^ cos 6 ^ — . - . 



z \ K z 



dx 



dx 

 dl 



dv 

 dy 



du 

 dx 



^irca a '2 TV a cos dx 



-TT . COS a 5- — . -— 



\z A. z dt 



27rco 

 2iv ca 



2'Ka 



+ -:r . COS ( 



Az 



dx 



' dt 



2'Ka a 



+ T^ . COS o . u. 



Now let 



then 



-ir- . COS f = — 



A 



u = b sin 6 ; 

 2^ ca 



\h 



n 2'7rab . n a 2'Kca^ . n a 



. COS + -^j— ; — . sm o COS f H — .^ ,., . sm o cos v ; 



\h 



\h^ 



but, from the hypothesis already made, the last two terms must be omitted ; 

 hence bh=ca- 



« = — . sin u 

 h 



27rca 

 V = -^^ — .y .COS. 



\h 



5. The hypothesis relative to the value of z, by means of which the preceding 

 results have been obtained, is that which belongs to the most simple case of wave- 



* PoissoN, art. 649 ; Moseley, art. 205 ; Pratt, art. 564 ; Webster, art. 108. The equation is 



dn dt dw 

 dx dy az 



