PROFESSOR KELLAND ON THE THEORY OF WAVES. 503 



quently -r- , which is equal to — ^ ,* is also independent ofy. 



X) will in consequence assume the form v=yf(x, t) + (p(x, t) ; but when y=0, 

 'c is always =0, hence v=yf{x,t) ; and, by reference to Art. 2, it appears that 



_ y da, 



a 



dt 



hence 



nr^ dy 



dy_ y da. 



dt a dt 



dz z da. 



dt a dt 



y dz 



z ■ dt 



1 dz _ 1 da. 



z dt a. dt 



Cor. 



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



dz 27rca ^ 2'Tra ^ dx 



-;— = ^ . cos f ;r — .cosf.— - 



dt \ \ dt 



y 2 TT c a ^ 2 TT a ?/ n dx 



V = — — cos a — -— — . - . cos o . — 



^^ A K z d t 



dv 2 ire a ^ 2ira cos 6 dx 



—- — —^ — . cos t; — -^- — . . — - 



dy Kz K z dt 



du 2'Kca ^ 21? a ^ dx 



^— = ;r . cos V ^ — ^ — . cos V . -— 



dx Kz Kz dt 



2ir c a ^ 2ira ^ 



= ;- . cos u + -^ — . cos V .u. 



Kz. Kz 



Now let u=b sin 6; 



,, , 2ir ^ 2ir ca ^ 2iTab . y. a 27rca2_ 



then — -^-- . cos V — ;r-- — . cos o + -^ . sin v cos v -\ — .. ,, ■ . sm a cos v ; 



A. Kh Kh K¥ 



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



hence bh=ca; 



ca . /, 

 u = — . sm f 

 h 



2irca ^ 



V = .y . COS. o . 



Kh 



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 



du dv dw 

 ax ay dz 



