160 MR. S. S. HOUGH ON THE APPLICATION OF HARMONIC 



already seen in Part I.* that the only forms of steady motion which can exist are of 

 this character, and have explained the fact by stating that the steady motions not 

 of zonal type which can exist on a globe without rotation must have their analogue 

 in the more general case in oscillatory motions whose period bears a finite ratio to 

 the rotation-period, no matter how great the latter may be. Our present work 

 confirms this statement and throws further light on the nature of these oscillatory 

 motions. 



.9. Evaluation of the Earlier Roots. 



The errors resulting from the use of the approximate formulae of the last section 

 may be considerable in the case of the earlier roots for which n has small values. 

 To obtain these earlier roots we must therefore proceed by trial and error, the 

 preceding method being made use of to obtain values with which to commence 

 the trials. 



As a concrete example we will discuss in detail the computation of the positive 

 root of the first class corresponding to the case n = 4, s = 1, when the depth is 

 given by %/4w 2 o 2 = -$. Taking p/cr = 0*18093, and introducing the numerical 

 values of n, s, and h, the equation 14 = becomes 



(X/u) 4 - 0-3333 (X/w) 3 - 5-5481 (X/co) 2 + 1-0906 (X/w) - 0'0418 = 0. 

 By HORNER'S process the greatest positive root of this equation is found to be 



2-43265. 



Now experience shows that the numerical value thus suggested is in general too 

 small. t We therefore select for a first trial a value rather larger than that 

 indicated, say, for example, 



X/w = 2-4400. 



From the formula (54) we find 



log a\ = 0-2553, log a\ = 1-1652, log a\ 17289, 

 log a] = 2-1450, log a\ = 2'4769, log"a = 27537, 

 log a\ = 2-9915, log aj = 3-2001, log al = 3'3859, 



while the values of the expression 



' 



for the values 2, 4, 6, 8, 10 of n are 



1-6046, 18794, 84-517, 250'92, 589'4. 



* H, 15. 



t Compare the 2nd and 3rd columns of Tables I. and II., Part I. 



