118 



PEOF. A. E. H. LOVE ON THE TRANSMISSION OF 



except for a change in sign, as the corresponding terms of H, while those which 

 contain the sine or cosine of (kCt + zO) are nearly the same and of the same sign. 



13. In order to sum the series denoted by S 11; ... it was necessary to evaluate R n 

 and tan X B . This can be done, when u n and v n are known, by means of the formulae 



tan XB = u n 



(W.' + V,*). 



(45) 



the second of these being obtained from (23). It was therefore necessary to calculate 

 u n and v n . Now, when q n is not more than a small fraction of (z)' /s , it is known from 

 the analysis of L. LORENZ already cited, and confirmed by MACDONALD and NICHOLSON, 

 that v n and u n are given very approximately by the equations 



. . . (46) 



' cos 2 

 1 'z 1 /' cos -J T sn , 



'' q n 



These results were used for the values |- and |- of q n , corresponding to the values 

 z and zl of n, but not for any numerically larger values of q n . For other values of 

 n the values of u n and <; were found from the sequence equations, deduced from (23), 



viz. : 



= 2n-l t _ 2/t -1 u / 4 *A 



n ^^ n 2 n 1) n * n 2 n 1" \ / 



For n increasing beyond z (q n > ^), these were used in the equivalent forms 



v n = 2v n _ l v n _2+ <A ~ -%'_,, u n = 2u n _ l -u n _. i + ^2 iM B _j, . (48) 



and, for n decreasing beyond 21, (g n < &), in the forms 



1 , (49) 



and the results were verified -each time by substituting in the formulae (47). 

 The wave-length taken was 5 km., so that ka, or z, was 8000. 

 The initial results thus found are given in the following table : 



TABLE I. 



