ANALYSIS TO THE DYNAMICAL THEORY OF THE TIDES. 
225 
and in general 
( - 
7.9.11 . . . (2?i + 5) 
1 
7.9’ 
1 
5.7’ 
0, 
1 
9.11’ 
0, 
0, . 
0 , 
0, , 
0 , 
1 
11.IJ 
0 
0 
, -L, 
'65 
15.17 
L 
0 , 
"■ 2 ’ {2n-l){2n+l) 
^ — L 
{2n-Z){2n-l) 
or, denoting the determinant which multiplies C 3 in the last equation written down 
by A„, 
a,3= (-1)'*/^7.9.11 . . . (2 w+ 5)A„C25 (n=2, 4, 6, . . . ) 
Now the determinant A„ is an algebraic polynomial of degree n/2 in If therefore 
we equate it to zero, we should obtain an algebraic equation of degree W./2. If 
has as its value any of the nj"! roots of this equation the first 7?/2 of equations (26) 
will be consistent, while 0,1 + 2 vanish. By increasing the value of n we shall 
approximate mor-e and more closely to the case where an infinite number of such 
equations are satisfied, while we shall impose an additional condition on the C’s, 
viz. :—that at some stage one of them must vanish. Though in general in the actual 
motion none of the quantities Cg, C^, . . . are zero, they are however subject 
to an important restriction, namely, that the series Cg, C 4 . . . must form a 
converging series, and therefore we must satisfy the equation 
B^C;, + 2 = 0 . 
n = 00 
The latter equation may be regarded as the period-equation for the free oscillations. 
It follows that as n is increased the roots of the equation A^^ = 0 , which make 
0 , must approach closer and closer to certain definite limiting values, which 
correspond to the different periods of free oscillation, and that the series Cg, C 4 . . . 
calculated in succession from equations (26) can only form a convergent series when 
/has one of these values. 
Now we see that A„ = 0 is the equation obtained by eliminating the C’s from the 
set of equations 
MDCCCXCVII.—A. 2 G 
