222 
MR. LUBBOCK ON THE TIDES. 
surface of the ocean from the earth’s centre, neglecting terms multiplied by the fourth 
power of the parallaxes, is proportional to 
3 m P 3 
2 
| cos 2 £ - 4} + 3 - m f - { COS 2 S' - Y j* 
where m is the mass of the luminary, P the horizontal parallax, and £ the zenith 
distance, the unaccented quantities referring to the sun, and the accented quantities 
to the moon. 
If a denote right ascension, S declination, l geographical latitude, and p sidereal 
time, 
cos £ = cos & cos / cos ((* — a) + sin & sin l. 
h, the height of the water, 
= - D ~ = W-( 1 - 4 cos*/) (1 - 3 Sinn) 
3 7 YI 7 ? 
-f — ~ cos2 ^ {cos 2 & cos (2 p — 2 ct) + 2 sin 2 & tan / cos {p — a)} 
m'RP' 3 3 , A „ 0 . 
- -tf- V 1 ~ cos l ) (1-3 sin 2 S') 
3 77 ?/ Z? plS 
-f- — — cos 2 1 {cos 2 cos (2 p — 2 a 1 ) + 2 sin 2 tan l cos (p — a!)}, 
M being the mass of the earth, and D a constant depending only on the zero line, 
from which the heights are reckoned. 
At high water 
mP 3 cos 2 ^ (n n h mP 3 sin2§tan£sin(/*. — a) 2 tan l tan S' sin (p — el) 
.../ 73/a ».M Sin [~ jU ~‘ a ) “ " 
tan (2 u — 2 a') = 
m 
- r -,\n( n z. n ^.') 
P 3 cos 2 S' ^ ' to' P' 3 costs' cos (2 ju. — 2 a') cos (2 /x — 2 a') 
1 + 
m P 3 cos 2 S 
m! P' 3 cos 2 S' 
cos (2 « — 2 a'). 
If — a' = ip, a — a! =z p 
m P 3 cos 2 8 
= ^ 
/A — CC ~ \f/ — <P 
m' P 13 cos 2 S' 
and if we neglect the difference in the interval for the morning and evening tides, 
A sin 2 <p 
tan 2 4> = 
l + A cos 2 <p ’ 
q, is the hour-angle of the moon at the time of high water, and is an angle differing'!' 
little from 0 or 180 °. 
3 m' R P 13 2/ 2 v, 
If E = -4^— cos 2 / COS 2 6, 
considering only the arguments 2 /a - 2a and 2 p — 2 a', 
* This amounts to supposing that the differential equation to the fluid surface is given by the equation 
Xd.r + Fdy-|-2’d^ = 0 
so as to neglect the quantity u' dx + v' dy + u>' dz in the notation of M. Poisson, Traitd de Mecanique, 
vol. ii. p. 6G9. 
t This has reference to the case of a perfect sphere. 
