Deep Water Waves. 391 



A 7 = 23'34a 7 4- 79'20a 9 + 498"3a 



11 



A 8 = 52-01a 8 + 207-4a 10 +139(V 2 , 

 -A 9 = 118'6a 9 +543'4a 11 , 

 A 10 = 275'6a 10 + 1426a 12 , 

 -A u = 649-8a u , 



A 12 - 1551a 12 , 

 _C = '5+a 2 + 2-75a 4 +13-92a 6 + 103'8a 8 +823-8a 10 , 



? 7r ^ 2 = l + a 2 +3-5a 4 +19-08a 6 + 154-7a 8 f 1297a 10 . 

 gX 



The series for rj certainly becomes divergent in the neigh- 

 bourhood of the crest of the wave when a is greater than 

 1/e, where e is the base of natural logarithms, and the 

 differential coefficients which occur in equation (3) become 

 divergent when a = l/<?. For the series formed by the 

 leadino- terms of the various coefficients is 



X 



i(m-l)I' 



and this is divergent when a is greater than 1/e, though it 

 converges when a — lje. For when m is large the mth term 

 of this series is 



m »i-2 a m e m-i! | ^/^rim - l)«-i 1 



= (ae) m m ~ v^/'Ztt, 



which proves the desired result. Further, the corresponding 



at] 

 term of the series for -jj. is of order (ae) m m~*, so that this 

 ag 



series diverges when a — l\e. Both series are, however, 



convergent when a<l/e. 



Moreover, the terms of any one coefficient are all of the 



same sign, so that the divergence of the series when a is 



greater than 1/e is increased by the presence of these terms, 



and its convergence when a is less than Ije is rendered 



doubtful. It is impossible, with the numbers given above, 



to say with any certainty whether the series for any given 



coefficient is convergent or not, but the general impression 



is that all become divergent when a = 1/3 or thereabouts. 



