ANALYSIS TO THE DYNAMICAL THEORY OP THE TIDES. 153 



The roots of the equation L' n = 0, regarded as an equation for the determination 

 of \/a>, will be the abscissae of the points of intersection of this curve with the line 



y = 



Since h is essentially positive, the roots will all be real, and they will lie in the 

 intervals between 



2s 2s 



' (+!)( + 2)' n (-!)' 



For large values of %,,/4<u 2 a 2 the two. extreme roots will approximate to the roots 

 of the equation 



% 



, X 3 n (n + 1) 2(0s/\ _ 

 """ - 3 "' 



~ 4w% 3 """ 4^ -n? (n +TJ 3 

 while the remaining two roots will approximate to 



___ 2s__ _2s 



(n + T) '(n + 2) ' n (n 1) ' 



The two former roots are those which have their analogue in the special case 

 treated of in Part I., for which s 0, in which case they have equal magnitudes but 

 opposite signs. These roots, we may expect, will approximate to roots of the period- 

 equation, at least when n is large. 



In order to see the significance of the remaining roots, it is convenient to transform 

 the period-equation into a different form, which moreover is far better adapted for 

 the more accurate numerical determination of the earlier roots. 



6. Modified Form of the Period -Equation. 



Eeferring back to the equations (29), (30), (32), which define x s ,,, y\, I/,, we see 

 that (31) may be written in the form 



(n S) | v , o if ~ 8 V"- J./ y'"-r^ p,, 



o N W-2T ./o,,, , 1\ ^ 



f X 3 n (n + 1) 2cos/\ hg n 1 _ 



. (M + 8+1) _ l"(/t + 2) 3 (M-S+ 1) c , (7t + S+2) c , "I _ 



Hence, if we introduce a new set of auxiliary constants, D], D* +1 , &c., such that 

 for values of n equal to or greater than s 



2n 

 VOL. CXCI. A. 



~^ - + !1I ^TT 1) c " '= {- C + - T} < 49 '- 



