ANALYSIS TO THE DYNAMICAL THEORY OF THE TIDES. 
227 
We may however obtain a number of alternative forms for our period-equation. 
From (27) we find 
1 
Cr^JCr _^ _ (2r - 3) (2r - l)'^ (2r + 1) 
(2r -f 3) (2r -f 5) 
(2r - 1) (2r + 1) 
and therefore by successive applications 
1 
1 
Cr + j 
(2r- 3) (2r - lf(2r + 1) 
5.7b 
(2r + 3) (2r + 6) ~ ' 
Ij)-_ 2 ““ • 
. . - L, 
Thus, we have 
1 
1 
(2,--3>(2,--l)»(2r+ 1) 
(2r + 1) (2r + 3f (2r -H 5) 
“ c,/a_2 + 
CrlGr + 2 
(2r - 1) (2r + 1) 
(2r + 1) (2r -f 3) 
1 
1 
1 
(2r - 3) (2r - l)^ (2r + 1) 
(2r - 7) (2r - of (2r - 3) 
5.7^ 9 
1 
I 
1 
1 
1 
- Lo 
1 
1 
1 
+ 
(2r-H)(2r + 3)3(2r-f5) {2r + 5)i2r + 7y {2r + 9) 
(271-3) (271-7^(271 + 1) 
Lr 
+ 2 
u 
+i 
This is an alternative form for the equation A„ = 0 ; by making n infinite, we 
obtain as an alternative form of the period-equation 
1 
L. 
(2r - 3) (2r - 1)^ (2r + 1) (2r - 7) (2r - 5f (2r - 3) 
5.7^ 9 
-‘r-z 
1 
1 
(2r + 1) (2r -H 3f (2r + 5) (2r5) (2r-t- 7)M2r-f 9) 
-^r + 2 
-‘r + i 
ad i7if. 
, = 0 (29), 
where r is any even integ’er. 
§ 7. Numerical Solution of the Period-Equation. 
The method I have used to solve the above equation will perhaps be best explained 
by giving a numerical example in detail. 
Taking for the ratio of the mean density of the earth to that of water the value 
given by Boys,* we deduce 
* ‘ Roy. Soc. Proc.,’ vol. 56, 1894, p. 132. 
2 G 2 
