AT SOUTHAMPTON AND AT IPSWICH. 49 
corresponding converted depressions, and to take their mean ; then those between 
5° and 10°; and so on. The phases thus obtained differed little from 2°*5, 7°’5, kc.; 
and the correction of converted depression for these small differences being easily 
found from the preceding and following numbers, the converted depressions for 2°-5, 
7°*5, &c. were found. In this manner the following Table was formed : — 
Corresponding phases and converted depressions in the Southampton Tides ; the 
phase commencing with low water and increasing by 360° in one tide ; and the 
depression being measured from high water, the whole range being called 2-000. 
Phase. 
Converted 
depression. 
Phase. 
Converted 
depression. 
Phase. 
Converted 
depression. 
Phase. 
Converted 
depression. 
O 
2-5 
1-996 
92- 5 
1-194 
182-5 
0-064 
27°2-5 
0-217 
7-5 
1-979 
97-5 
1-185 
187-5 
0-029 
277-5 
0-251 
12-5 
1-939 
102-5 
1-182 
192-5 
0-018 
282-5 
0-316 
17-5 
1-890 
107-5 
1-181 
197-5 
0-027 
287-5 
0-384 
22-5 
1-838 
112-5 
1-163 
202-5 
0-051 
292-5 
0-508 
27-5 
1-770 
117-5 
1-128 
207-5 
0-079 
297-5 
0-598 
32-5 
1-698 
122-5 
1-080 
212-5 
0-104 
302-5 
0-755 
37-5 
1-639 
127-5 
1-018 
217-5 
0-132 
307-5 
0-890 
42-5 
1-574 
132-5 
0-941 
222-5 
0-145 
312*5 
1-021 
47-5 
1-519 
137-5 
0-846 
227-5 
0-149 
317'5 
1-201 
52-5 
1-479 
142-5 
0-748 
232-5 
0-152 
322-5 
1-347 
57-5 
1-422 
147-5 
0-651 
237-5 
0-144 
327-5 
1-491 
62-5 
1-382 
152-5 
0-563 
242-5 
0-139 
332-5 
1-641 
67-5 
1*329 
157-5 
0-441 
247-5 
0-125 
337-5 
1-769 
72*5 
1-292 
162-5 
0-350 
252-5 
0-129 
342-5 
1-864 
77-5 
1-256 
167-5 
0-262 
257'5 
0-132 
347-5 
1-932 
82-5 
1-225 
172-5 
0-185 
262-5 
0-143 
352-5 
1-976 
87*5 
1-209 
177-5 
0-119 
267-5 
0-174 
357-5 
1-996 
By means of these numbers the curve fig. 1 in Plate II. has been constructed. 
To express these numbers in a mathematical form by means of a periodic function, 
it was assumed that the converted depression could be represented by the formula 
A + B . sin phase + C . sin 2 phase + D . sin 3 phase + E , sin 4 phase, 
+ b . cos phase + c . cos 2 phase + d . cos 3 phase + e . cos 4 phase. 
Then A is the mean of all the converted depressions. B is found by multiplying 
every converted depression by its value of sin phase, and dividing the sum of all the 
products by 36. b is found in the same way, using the values of cos phase as factors. 
C is found by using sin 2 phase as factor, &c. In this manner the following expres- 
sion was found for the converted depression : — 
0"900 + 0-469 . sin phase — 0 - 085 .sin 2 phase — 0*057 • sin 3 phase — O'0 17 . sin 4 phase 
+ 0'766 . cos phase + 0-174. cos2 phase + 0- 1 82 . cos 3 phase — 0*029 . cos 4 phase ; 
or 
0'900 + 0-898 . sin (phase + 58° 30') 
+ 0-194 . sin (2 phase + 116°) 
+ 0*191 . sin (3 phase + 107° 22') 
— 0-034 . sin (4 phase + 59° 43'). 
H 
MDCCCXLIII. 
