OF SMALL DEPTH AND WIDTH. 



459 



Now if 



y = ± (i. 



represent the equation of the two sides of the canal, we need only 

 satisfy one of them as 



y - /3. = 0, 



since the other will then be satisfied by the exclusion of the odd 

 powers of y from (p. 



The equation (A) gives, since here A = y — (i 



Similarly, if x - 7^ = is the equation of the bottom of the canal, 



If moreover, x — '(^ , = be the equation of the upper surface, 



(c). 



dz dx dx dt 



But here^j = ; .-. also by (2) gX^ = 



d^ 

 ~dt) 



(when s; = ^). 



Substituting from (3) in (b) we get 



or neglecting quantities of the order 7°, 



(i'). 



„ , , d^ d(p„ 



Similarly (a) becomes 



» = *''^-£t- 



and (c) becomes, since ^ is of the order of the disturbance, 



W). 



-*.-'l 



y d(t>„ 

 S^=dt) 



when a = ^, 

 or neglecting (disturbance)" ss 



=o}. 



provided as above we neglect (disturbance)', 

 Vol. VI. Pabt III. 3 N 



