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



Thus the equation which involves y' n becomes 



whence 



01 = 



__. - 14 + K' n+2 

 We may now deduce C^_ 2 , C*,_ 4 , . . . C', +2) C' [+4 , ... by means "of the formulae 



C" TT P 



^n-2 "-! ^a-t 



-- - i 



' 



o 



and therefore 



_, _ 2n _. 



- 4V H' a _ a - L; + K; +2 [; ' ;4- 2 *?,- 



TJ K" tr TT-. -i 



^ ^ JVIV - p, . 



- ^>.+4 4- 

 J 



It should 'be noticed that the term involving P^ need not here be the predomi- 

 nating term of the series within the square brackets ; it will however be so when 

 the value of X for the disturbing force is in the neighbourhood of those roots of 

 the period-equation which approximate to roots of Lj, = 0. But if X have as its 

 value another root of the period-equation, say, for example, one which approxi- 

 mates to a root of L?,, = 0, the series within square brackets will differ only by a 

 constant multiplier from the series in the expression for the tide-height for the 

 corresponding- type of free oscillation, since the equations which determine the ratios 

 of the C's are evidently the same in both cases. Hence, for values of X in the neigh- 

 bourhood of this one, the predominating term will be that which involves Pf^. 

 Consequently, when n and m differ widely, the numerical computation of these 

 series will become laborious ; for as we proceed away from the term containing 

 'P towards that involving P;;,, the terms will at first increase in magnitude, and 

 the convergence of the series will not assert itself until the term depending on 

 P has been passed. These circumstances will not however occur in any of the 

 cases of more practical interest. 



Whatever be the nature of the disturbing potential it will be possible to expand 

 its surface-value in a series of surface-harmonics. Thus the most general value 

 of v which can occur may be expressed in the form 



=o = 



The deformation of the surface due to each term may be calculated independently 



