HARMONIC ANALYSIS AND PREDICTION OF TIDES 99 
have a method of using these means directly in the analysis for the 
annual and semiannual constituents, thus avoiding any special sum- 
mation for the purpose. The period of the annual constituent is ap- 
proximately the length of the Julian year, that is, 365.25 days. If this 
period is divided into 12 equal groups and the mean of the hourly 
heights for each group taken, these means represent the approximate 
height of the combined annual and semiannual constituents for the 
middle of each group, and the middle of the first group will be the 
initial point from which the zeta (¢) as obtained by the usual process 
is referred. As each group represents 30° of motion for the annual 
constituent, or 60° for the semiannual constituent, to refer this ¢ to 
the actual beginning of the series of observations it will be necessary 
to apply a correction of 15° for the annual constituent or 30° for the 
semiannual constituent. 
281. In obtaining the monthly means by calendar months the year 
is divided only approximately into 12 equal groups. The following 
table shows the difference between the middle of each group repre- 
senting a calendar month and the middle of the corresponding group 
obtained by dividing the Julian year into 12 equal parts. It is to be 
noted that the hourly heights included in a monthly sum extend from 
0 hour on the first day of the month to the 23d hour on the last day. 
The middle of the group as reckoned from the beginning of the month 
will therefore be 13.98 days, 14.48 days, 14.98 days, or 15.48 days, 
respectively, according to whether the month has 28, 29, 30, or 31 days. 
Middle of group reckoned 
from beginning of year Differences 
Month ee 
Julian |Common| Leap }Common| Leap 
year year year year year 
Days Days Days Days 
15. 48 15. 48 +0. 26 +0. 26 
44,98 45. 48 —0. 68 —0.18 
74. 48 75. 48 —1.61 —0. 61 
104. 98 105. 98 —1.55 —0. 55 
135. 48 136. 48 —1.49 —0. 49 
165. 98 166. 98 —1. 43 —0. 43 
196. 48 197. 48 —1. 36 —0. 36 
227. 48 228. 48 —0. 80 +0. 20 
257. 98 258. 98 —0. 74 +0. 26 
288. 45 289. 48 —0. 68 +0. 32 
318. 98 319. 98 —0. 61 +40. 39 
349. 48 350. 48 —0. 55 +0. 45 
aS be 8 AR le Sp a —11. 24 —0. 74 
Bree Se | ew ee —0. 94 —0. 06 
Speed of Sa constituent per day=0.9856°. © 2 
Mean differences reduced to degrees of Sa________________________-_-_-___-_-_-_---_-- —0. 93 —0. 06 
Gorrechiontorgot¢saewr ae etree ee eC lh et eee Le eee 14. 07 14. 94 
Correction to ¢ of Ssa_______> Se reer ae ae ertee eee ae NE On See 28. 14 29. 88 
282. From the above table it is evident that in the summation for the 
monthly means for a calendar year the middle of each group of a 
common vear is on an average 0.93° earlier than the middle of the 
corresponding group when the Julian year is equally subdivided 
and the middle of each group of a leap year is on an average 0.06° 
earlier. Subtracting these values from 15°, the interval between the 
beginning of the observations and the middle of the first group of an 
equal subdivision, we have 14.07° and 14.94°, for common and leap 
years, respectively, as a correction to be applied to the ¢ of Sa as 
