4 
MR. AIRY ON THE LAWS OF THE RISE AND 
If we express the numbers in these two columns by series of sines and cosines of 
multiple arcs, we find the following expressions : — ■ 
Division A (neap tides). 
Converted depression = 1*035 
+ 0*198 sine phase — 0'918 cos phase 
+ 0-064 sine 2 phase — 0-016 cos 2 phase 
— 0"010 sine 3 phase — 0-047 cos 3 phase 
+ 0'008 sine 4 phase — 0-009 cos 4 phase 
= 1*035 — 0*939 cos (phase + 12° 10') + 0-066 sine (2 phase — 14° 2') 
— 0*048 cos (3 phase — 12° l') + 0*012 sine (4 phase — 48° 22'). 
If we make phase + 12° 10' = p + 90°, and if we remark that, to obtain the actual 
depression in feet and inches below the top of the wharf-wall, we must multiply the 
converted depression by half the range, and must add to the product the depression 
of high water, we find for the actual depression, 
15 ft. 3 in. f 1 -035 + 0-939 . sine (p) - 0*066 . sine (2 p — 38° 22') 
5 ft. 11 in. + 5 Xs 
2 L - 0-048 sine (3 p - 48° 40') + 0*0 12 sine (4 p- 97° 2') 
or 
13 ft. 10 in. + 7 ft. 7*5 in. X 
0 - 939 sine (p) — 0*066 sine (2 p — 38° 22') 
- 0-048 sine (3 p— 48° 40') +0*0 1 2 sine (4 p — 97° 2') 
where p represents an angle increasing uniformly with the time and going through a 
change of 360° in one complete tide. 
Division B (spring tides). 
Converted depression = 1*017 
+ 0 - 057 sine phase — 0-900 cosine phase 
+ 0*104 sine 2 phase — 0*022 cosine 2 phase 
— 0-054 sine 3 phase — 0*043 cosine 3 phase 
+ 0-016 sine 4 phase — 0*029 cosine 4 phase 
= 1-017 — 0-902 cosine (phase + 3° 37') + 0*106 sine (2 phase — 11° 57') 
— 0*069 . cosine (3 phase — 51° 28') + 0-033 sine (4 phase — 61° 7'). 
If we make phase + 3° 37' = p + 90°, we find as before for the actual depression 
of the water below the top of the wharf- wall, 
3 ft. 6 in. + 
19 ft. 2 in. 
2 
X 
1-017 + 0-902 . sine (p) — 0-106 sine (2 p — 19° 1 1') 1 
— 0*069 sine (3 p — 62° 19') +0-033 sine (4 p — 75° 35') j 
oi- 
l.S ft. 3 in. + 9ft.7in. X 
0*902 . sine ( p) — 0’106 sine (2p — 19° 1 1') 
1 
- 0-069 sine (3p - 62° 19') + 0 033 sine (4 p - 75°35')/’ 
where (as before) p represents an angle increasing uniformly with the time, and going 
through a change of 360° in one complete tide. 
