108 Mr. R. F. Gwyther on Progressive 



between and 6 2} the solution requires 



#1 



From the quadratic (19) we also have 

 1 + d 2 =l-3ct, 

 6 } d 2 =S* 2 -2a, 

 from which, by elimination of a, we get 



*=^ri < 21 > 



Als0 3/* = l -\(2-* 2 ) (22) 



In the same way, for any selected interval between the 

 roots we arrive at the same relations (21) and (22). The 

 correctness of the result can also be verified by differentiating 

 (8) and reverting to the equation 



y-gh/Z)hy>'=-3cf' + 2(c*-gh)f, . , (23) 

 and substituting in this equation the value of / given by 

 putting y = Q in (20). 



We thus obtain the condition 



X 2 (3dn 4 u - 2 (2 - « 2 )dn^ + 1 - * 2 ) 

 = 3 (Xdift* + /juf - 2(\dn 2 w + p) , 



where u stands for 



\{c*—gh) x 

 2(c*-gh/3)'h 



On comparison of coefficients we again find the relations (21) 

 and (22), and the solution is given by (20), (21), and (22). 



I wish especially to call attention to the point that the sign 

 of the radical in (20) must be positive, and that we get no 

 wave of depression by this method. 



These equations show that for any value of k between zero 

 and unity, we have numerically equal, positive and negative, 

 values of A,. On referring to (20), it will be seen that the 

 positive value of \ requires that c 2 >gh, while the negative 

 value requires gh>c 2 . We have thus two waves with 

 different rates for every value of the modulus, and, from 

 (22), these will have also different values for fi. 



These waves will, in general, be necessarily accompanied by 

 a motion of drift, for if we remove the purely periodic part 

 from the right-hand side of (20) we shall remain with 



/ Mc*~ 

 V 2(c 2 -, 



