102 PROFESSOR KELLAND ON THE THEORY OF WAVES. 



ct) = b(e Ky + e- Ky )sm6 . (1) 

 -ct') = -l>(e* y -e- Ky )cos6 (2) 



K *-e- a *)sin0 ...... (3) 



The equations of motion are (Art. 8.), 



We proceed anew to the discussion of these equations, reserving only the 

 solution of the problem given in Art. 8. 



By (4) and (5), _ .... (7) ; 



the differential coefficients which have the symbols enclosed in brackets being 

 complete, the others partial. 



d fdu du du\ d fdv dv dv\ 



Hence ^-[-rr + u ^ + w TT~ ) =^- ( ~r; + M 3- + ^- ) 



dy\dt dx dy) dx\dt dx dy) 



d du d du . 



But -j- -r=-r f -3-, &c. = &c. 



dy at at ay 



d (du .du du\ 



3-lTT + M ^~ + v ^r~ I may be written 



dy\dt d x dy) 

 d du d 2 u du du d 2 u du. dv 



- . - + U -- 1 -- . -- h V -- 1 -- - . 



dt dy dxdy dx dy d y 2 dy* dy 

 . ., T ... d fdv dv dv\ 



Similarly, the quantity ^ (j- ( + u + v j may be wntten 



d dv d 2 v du dv d 2 v dv dv 



dt ' dx da? dx ' dx dxdy dx ' dy' 



XT / d\du d du d du d du 



OW I -7-. ) -j-=-n ^- + T--J- M +^--^-"' which is less than the quantity 



\dtj dy dt dy dx dy dy dy J 



i v d fdu\ . du du du dv du fdu dv\ 



which represents ^) by ^ - Ty + J - y . ^ or by j- y . (^ + ^). 



' ; But ^^=0 by (6) therefore the quantities l(^) and (^)g 

 incident. 



