Waves in Flowing Water. 523 



of this problem. It is founded on a well-known analytical 

 method of Cauchy's, of which examples are given in the 

 Eighteenth note (p. 284) to his Memoir on the Theory of 

 Waves*. 



First, bring the denominator of (23) to the form of the 

 product of an infinite number of quadratic factors, as follows: — 

 Let 



W= — * {«* + «--£(«*-<-.)} . (24). 



2 V~ b) 



Expanding in powers of cr, we have 



b 



+ T^374( 1 --5?) ff4+&c -} • <»>■ 



Hence, when b is greater than D, W is positive for all real 

 values of cr. But when b has any positive value less than D, 

 W (which is always positive for small values of cr 2 ) is nega- 

 tive for large values of cr 2 ; and therefore at least one positive 

 value of a 2 makes W zero. We shall see presently that only 

 one positive value of a 2 does so. We shall see that all the 

 zeros of W when b is less than D, and all but one when b is 

 greater than D, correspond to real negative values of cr 2 . 

 This indeed is obvious if for a 2 we put — 2 , which gives 



w * ( cos ,_^ } . . (26); 



b 



and which shows that the zeros of W are given by the roots 

 of the well-known transcendental equation 



When b is greater than D this equation has all its roots real, 

 and in the first, third, fifth, &c. quadrants. When b is less 

 than D the root in the first quadrant is lost, and in its stead 

 we clearly have a pure imaginary ; while the roots in the 

 third, fifth, &c. quadrants remain real. Let 6 ly 2j #3? &c. be 

 the roots of the first, third, fifth, &c. quadrants. As the first 

 term of equation (25) is unity, we have 



* Memoires de VAcademie Roy ale de V Institut de France, savans 

 Hrangers, tome i. (1827). 



