OF SMALL DEPTH AND WIDTH. 227 



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

 powers of y from </>. 



The equation (A) gives, since here A = y /3, 



Similarly, if z y x = is the equation of the bottom of the 

 canal, 



A dd> dy d(f> . . 



Q = -- ....... ( when * = 7) ............. (J). 



If moreover z ,., = be the equation of the upper surface, 



,<*_##_# 



dz dx dx dt 

 , 



But here^ = ; /. also by (2) #? = ^ 



dt 



Substituting from (3) in (b) we get 



or neglecting quantities of the order 



-* 



Similarly (a) becomes 



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



-*-s 



or neglecting (disturbance) 2 z = j " 



provided as above we neglect (disturbance) 2 . 



Again, the condition (2) gives by equating separately the 

 coefficients of powers and products of y and z, 



= ^ + ^ + 6 "I 



dx' ** I , 



= &c. J 



152 



