ANALYSIS TO THE DYNAMICAL THEORY OF THE TIDES. 
241 
§ 12 . Lunar-Fortnightly Tides in an Ocean of Variable Dejjth, 
A similar method of treatment may be employed when the depth follows the less 
restricted law, of the form k I {1 — y?), made use of in § 5. The numerical com- 
m -:— where r is 
(r + VUr 
a small integer, since, as we have seen, the series which express the tide-heights will 
then rapidly terminate. 
For example, taking r = 4 so that 
putation is greatly facilitated in this case when I takes the for 
0 
4<y"-fd 
J._ 
• 5yjy 
which makes the value of I for the earth about 15,454 feet, the values of Cg, C 4 arc 
given by the equations 
+ 7 73 
y TP _ ^72 _ 
4:2^2 5.7 4a,kd“ 
whilst all the remaining C’s vanish. The notation employed is that introduced at 
the end of § 5. 
Taking p/o" = T8093, we find in the case of the lunar-fortnightly tides 
L 3 = - -27060, L^. = - -23787, 
log 4 = 2-3105, 
whence we obtain at once 
^ 0 \ ff2 % 
C, = - fA-. + ^ ^ = •74266,, 
\4a)^fd 7 4:(o~a^J (j V 
= O 4&» fl + ^ “ '09806^. 
1 t 
Thus, if o = -^ -h 7 - 7 — sin® 6, where 6 denotes the co-latitude, which for a 
5 4.5 
system of the dimensions of the earth makes the law of deptli 
(58080 + 15454 sin® 0) feet, 
we find for the tide-height the expression 
{ e3[-7426P2 - -0980PJ. 
2 I 
MDCCCNCVH.—A. 
