ANALYSIS TO TEE DYNAMICAL THEORY OP THE TIDES. 
245 
whence finally 
C, = -0092, Cg = *00023, Cjo = *00001 ; 
and the expression for the tide-height becomes 
[• 4719 P 3 - *1852?^ -f •0092P6 + •00023P8 + 'OOOOlPio + • . .]• 
As a further illustration, I have computed the series for the tide-height for the 
case where ^ — sin^ 6^, that is, where the depth is 14,520 feet at the 
poles and shallows to 4840 feet at the equator. The value of ^ in this case is as 
follows :— 
^ = ^2['3082P3 - -llOfiP^ -I- •0467P6 - •0158P8 -h •0048Pio 
- -OOMPig H- -OOOlPi^ - -OOOlPjfi -f . . .]. 
When I is positive, that is, when the depth at the equator exceeds that at the 
poles, the series appear to converge more rapidly than when the depth is uniform, 
but the opposite is the case when the water is deeper at the poles than at the 
equator. 
§ 13. Forced Oscillations of Infinitely Long Period* 
If we suppose X so small that we may neglect we find, on putting = 0, 
for the height of the forced tides the following four series in place of those given 
in S 11 
._L 
^o/Pj 
_L. 
WPa 
•266IP2 - •I671P4 + • 0482 P 6 
“ 4070 P 3 - -leeeP^ + •0284P(5 
• 5689 P 3 - • 1385 P 4 . + •0130Pfi 
• 7201 P 3 ~ • 0973 P 4 -h •0048Po 
- • 0080 P 8 + •OOO9P10 - -OOOlPia + . . . 
- •0026P8 + •OOO 2 P 10 - . . . 
- •0006P8 + . . . 
- • 0001 P 8 + • • • 
The lunar-fortnightly tides therefore differ only very slightly from tides whose 
period is infinitely long. The difference between these latter and the solar semi- 
* Several o£ the conclusions of the present section have been previously arrived at by Professor 
Lamb (‘ Hydrodynamics,’ chapter viii.); but, on account of the important light which they throw on the 
later sections, I have thought it desirable to treat the questions in some detail, even at the risk o 
repeating what is already well known. 
