Long Waves in Canals and Standing Waves in Closed Basins 171 



Equation (VI. 62) then gives m+ 1 simultaneous, homogeneous equations. 

 ^-■U) A + I ---M) A,+ I— -J,A 



They can only exist when their determinant vanishes: 



1 1 1 



Jn A — — •/ 1 a y> a 



3 6 10 



1 2 1 



-y,A -j,x — / 3 a 



6 15 10 



1 1 3 



JaA J 'lA J A A 



10 10 35 



in which 



'--/ 



z 2 (l-z 2 )z" 

 o{z) 



dz . 



(VI. 63) 



In case that in (VI. 61) A only consists of three terms {m = 2), we get A = z(l — z)(A J rA l z+A 2 z 2 ) 

 and the determinant is restricted to the terms shown in (VI. 63). From this we obtain as a determin- 

 ing equation for A the cubic equation: 



3,1 2 _ 1 



A 2 + 



+ 



1 3 53 1 1 

 700 350 2100 30 60 



A- 



I 



10500 



The roots of this equation must all be positive and can be determined 

 by applying the "regula falsi" or the method of Newton. If they are A 1? a 2 

 and A 3 , the second of the equations (VI. 58) gives as period of the uni-nodal, 

 two-nodal and three-nodal seiches the values 



T t 



Ina 



7&g) 



(VI. 64) 



Hidaka has applied his method to Lake Yamanaka, which has a beautiful, 

 regular shape and has obtained results which agree exceptionally well with 

 the observed periods of this lake. Neumann has applied this theory to the 

 large, almost completely closed water-masses of the Baltic Sea. We will refer 

 later to this important paper (p. 194). 



