Waves in Flowing Water. 525 



and 



sin«,[(i— |)<7r— a<] = D/b .cos ot { . . . (35); 



or, as is convenient for approximation, when i is very large, 



*[(t-i)w-^]»Dy&.j2- .... (36), 



which shows that as z increased to infinity, the value of a 4 

 approaches asymptotically to D/b (i — £)7T. Hence when fc" is 

 very large, the second member of (36) becomes approximately 

 D/b . (1 — 3 «i) ; and the equation becomes 



ll- i J)lb)^-(i-iyu i =-I>lb . . (37); 



a quadratic, of which the smaller root when D is less than 3 b, 

 and the positive root when D is greater than 3b, is the required 

 value of a*. 



Going back now to (23) and modifying it by (24) and (29), 

 we have 



X(T 



*-r^- s *«J>7l? (38); 



or, according to the well-known evaluation (attributed by 

 Cauchy to Laplace) of the definite integral indicated, 



B - *A/D 



.tdJSie-£ (39); 



D/b 



or with i} Nj eliminated by (33) and (34), 



N4A/D.S ( /J^^ « » . . (40), 



where a 1} « 2 > • • • a i denote all the positive roots of (35). 



This series converges with exceeding rapidity when # i3 

 any thing greater than D, and with very convenient rapidity 

 for calculation when x is even as small as a tenth of D. 

 When # = 0, the convergence has the same order as that of 

 1— e + e 2 — &c, when e=l; and we find the sum by taking 

 as remainder half the term after the last term included. 

 The true value of the sum is intermediate between the values 

 which we obtain by this rule for a certain number of terms, 

 and then for one term more. When it is desired to obtain 

 the result with considerable accuracy, a large number of terms 

 would be required; and it will no doubt be preferable to use 

 my first method as indicated above. 



It remains to deal with the first term for the case D > b, 

 which makes it imaginary in the form (39), but real in the 

 form (38) with — <7 X 2 substituted for 0, 2 . For this case we 



