DKTKCTIOX OF TWO MOIH'LATEP WAVES 19 



the case of a term in cos pi derived from term III of (5). This term is 

 sHghtly larger than the above Hmit when M = 0.5 and e/E = 0.1 but 

 as it decreases rapidK with a decrease in e E it has been omitted for the 

 sake of simpHcity. 



Having chosen this Hmit of 0.005 for the relative magnitude of 

 individual terms it can be shown to be permissible to neglect term IV 

 and all subsequent terms of (5). Furthermore, only a few of the large 

 number of terms yielded by III need be retained. 



After applying these rules there appear several frequencies that are 

 never as large as 0.01 in relative magnitude and these ha\e been 

 omitted from consideration. As has been stated in the body of the 

 paper, an exception is made in the case of the frequencies {p±q±ii)/2ir. 

 If a given frequency exceeds 0.01 for any one of the four pairs of 

 values of .1/ and /;/, it has been shown on* the figures for all of the 

 pairs. 



After the formula (5a) has been applied to 6' and the expressions for 

 A and B inserted there remains the necessity of reducing products and 

 powers of various sinusoidal terms to sums of simple first order 

 sinusoids. This is a tedious procedure but is a matter of simple 

 trigonometry and will not be set forth in detail. 



From (5a) it can be seen that if M or m is near unity the series will 

 converge very slowh. Furthermore, since to obtain relative magni- 

 tudes we divide by M, it is impossible to obtain satisfactory convergence 

 due to small values of M in the denominator. Hence it is necessary to 

 limit M and m to 0.5 or less and in addition .1/ must be no smaller than 

 0.1. It would be permissible to allow m to become less than 0.1 but as 

 little would be gained by this ;;/ has been restricted to the same range 

 as AL 



