﻿Wave of Permanent Type. 355 



.*. (19) gives when a- is small, 



o-=-Bmr/5A, (20) 



as a convenient approximation for the form of the surface in 

 the neighbourhood of the crest. 

 Again, (11) may be written 



hp=\p{\P- q *)-gp{c-h-t;), .... (21) 



whence, since q and £ vanish together, to make 8p vanish we 

 must have 



W=2 9 (c-h), (22) 



and this reduces (21) to 



Sp=ffpS-W (23) 



Now from (16) we obtain 



q 2 /TJ 2 pmr{A 2 + 2ABmS+&c.}, . . . (24) 

 and thus 



8p= P r{g cos $-imA 2 pU 2 + ABU 2 mZ+&c.}. . (25) 



Hence, since &= -x + <r we must have, to make 8p vanish to 

 o 



a first approximation, 



gcos ^ — ^mA 2 pU 2 =0, 



that is 



\] 2 =g/mA 2 p, (26) 



and (25) becomes 



g^U^ -|(l- yUBmV + &c. . . (27) 



This cannot vanish unless B vanishes, in which case we see 

 by (20) that the curvature of the surface vanishes close to 

 the crest. This result is obviously independent of our ap- 

 proximation, but we have not taken enough of constants to 

 secure it here : it will be found, however (v. § 5), that the 

 other equations determining the constants we have at our 

 disposal will make B very approximately vanish. 



4. Numerical Determination of tlie Constants. 



There is still one important condition to be satisfied. To 

 ensure the connexion between our separate treatment of the 

 neighbourhoods of the crest and mean level, we must secure 

 that the stream-line a/t = — JJh, bounding the distant surface, 

 shall pass through the crest : or, in other words, the flow 

 across any infinitely distant section must be equal to the flow 



2B2 



