8 GWYTHER, Range of Progressive Waves in Deep Water. 



The Reversal of the Series. 



The next stage in the process is to shew how to 

 express ^, and finally x and y, explicitly in terms of ju, 

 and therefore in terms of t. The whole of the results thus 

 obtained as far as (20), are generally applicable in cases 

 of wave motion, and are independent of the special values 

 of the constants which are introduced at the next stage. 



It will be noted that the relation (8) is of the same 

 character, though of course more complex, as that con- 

 necting the mean and the eccentric anomaly, and that it 

 is capable of reversal by the method which Bessel applied 

 to the astronomical problem. The same method may 

 also be applied to express x and y in terms of /. 



The relation (8) is 



^ + iS'jSin(/» + -£'2sin 2^ + ... =;U, 



hence when = ^Tr we have ^ = r-K, and we may properly 



assume 



= y^ + ^jSinju + ^„sin 2/i+ ... (9). 



Differentiating this last equation with respect to ju, 

 multiplying throughout by cosr/^ (where r is any positive 

 integer), and then integrating with respect to ju between 

 the limits and tt of ju, we remain with only the term 

 arising from sin r\i in the series in (7). 



Accordingly we obtain 







—COS rfidfi, 

 dfx 



and since, as we have noted, has the value rir when fx 

 has the value rw, it follows that we may change the 

 independent variable from fi to ^, and retain the limits 

 unchanged. 



