On Progressive Long Waves in- Shallow Water. 107 



and writing f'=a + 0, we should get 



(c 2 -ghfi)K*6* = - 2c6(0 2 + %*Q + 3a 2 ) . 



The solution will introduce a term of the form tan 2 -£, 



where <$> is the amplitude of an elliptic function. This 

 solution will be rejected, as not representing a wave-motion 

 such as we seek. 



When c 2 =fcgh, we write f— — — — %, and obtain from (8) 



1 AY 2 =-2( % W) + 2( X W). 



c 2 -^/3 



Further, putting ^=a+0, we get 



_^r^_7 i 2^ = __26'(^ + (3a-l)6> + (3«-2>). (18) 



The different possible solutions depend upon the roots of 

 the quadratic 



2 +(3«-l)0+(3a-2> = O. . . . (19) 



These roots are imaginary, unless a lies between the values 

 — ^ and 1, and the solutions corresponding to these imaginary 



c6 

 roots we reject, because they all introduce the form tan 2 ^, as 



in the previous case. 



When the roots are real, we shall have not only solutions 

 corresponding to intervals between the roots, but also corre- 

 sponding to intervals between the extreme roots and positive 

 and negative infinite values. 



These last solutions, introducing sec (f> and cosec </>, must 

 also be rejected. 



We shall then retain only solutions suitable for periodic 

 waves, corresponding to intervals between the roots. These 

 solutions all present themselves in the same form in terms of 

 the elliptic function of the second kind, and the special cases 

 Corresponding to coincidences of roots are easily treated. 



The form of solution which continually presents itself is 



j., . . c 2 —gh . . 



If #! and 2 are the roots of the quadratic (19), and if we 

 suppose #x and 2 both positive, and consider the interval 



