PROFESSOR KELLAND ON THE THEORY OF WAVES. J43 



therefore in the latter case 



in the former 



< t >= ^\2 + 2) = c as it evidently ought to be. Hence it appears that the instan- 



taneous impulse conveyed to any point in the surface of the fluid is exceedingly 

 small. 



2. If y=-h 



sin /i (a -* + *) + sin //(*-*) e~* h 



But 



_2C r a sin /J. (a x + k~) + sin fJL (x k) ph,, /* -^. . - 



" 



_i xk \ a x + k \ xk \ ax+k 



tan r + tan -. tan -rrr- tan _ 



A A on on 



l xk i a x + k s 



tan -=-,-- + tan - -. &c. 



5 A 5 A 



*> r ( * i- i i < &\ 3 1 /r t 

 _*^ } r ~ K x ( f ~\ + i(f l: 



TT I A :> V A / 5 \ A 



_ _ a j? + 



" ~~ + ~ 



-A 1 /x k\ 3 I/x k\ 5 

 "3A + 3V3A j -51-3TJ 



+ &c. &c. 

 2Cra la la 



_____ -i ____ 



" -TT \ A 3 A 5 A 



- - 



3 W 33 3 A; 3 5 3 \A 



+ &c. &c. | 

 = -T nearly; if xk be small and A considerable. Hence we 



IT h 4 



learn that the impulse instantaneously communicated to the bottom of the canal 

 varies inversely as the depth. 



The few cases we have exhibited above, must not be supposed to include all 

 that our analysis is capable of developing. We have given these cases rather 



