OCEAN IEMFESATVBE8 371 



substituting this value of a^ in equation (89) gives 



&i = !L-== (95) 



2 



which agrees with the result already found on page 370, that -^ 



PI 

 must be negative. From pages 369 and 370, 0Me av +, where 



a aizt&i* and b = a 1 = (2a lf j. 2 wjb^ 



The solution of equation (81) is therefore 



= j|f e iv (fciiH-aie)i (96) 



where M and a t are arbitrary constants, a : and &! are given by equa- 

 tions (94) and (95) and the same value of a x is to be used with either 

 the plus or the minus sign before the expression in brackets. From 

 the properties of imaginary exponents equation (96) can be put in 

 the real periodic form 



$=0*{A l mn(l> l <y + a 1 t)+Bieo* (fc^ + o^)} (97) 



where A^ B 1 and a t are arbitrary constants. 



Also since the differential equation is linear the sum of any number 

 of such expressions will be a solution. Therefore the following more 

 general expression 



n = 00 

 0= 



nv I A n sin (b n y + a n t)+B n cos (b n y + a n t) j (98) 



is a solution, where A n , B n and a n are arbitrary constants and a n and 

 b n have the values 



and 



V2~a n 



Denoting - - \/-^f- by A,, and l by A the following approximate 

 expression for a n and & can be easily derived 



a =4+ (1 + *n 2 + An 4 ) A,, . (101) 



6=(1 A n 2 7 W 4 )A (102) 



