92 U. S$. COAST AND GEODETIC SURVEY 
266. The complete augmenting factor for the long-period constit- 
uents, the year or month being represented by 24 means, will be 
obtained by combining the above factor with that given in formula 
(308). Thus 
Tp 24 sin 3a 
Se =X 
pain “EP sin 12a 
augmenting factor= (403) 
If the year or month is represented by only 12 means as when monthly 
means are used in evaluating Sa and Ssa, the formula becomes 
Lap 24 sin 3a 
12 sin 15p”~ sin 12a 
augmenting factor= (404) 
Values obtained from these formulas are given in table 20. 
267. The following method of reducing the long-period tides, which 
conforms to the system outlined by Sir George H. Darwin, differs to 
some extent from that just described. In this discussion it is assumed 
that a series of 365 days is used. Let the entire tide due to the five 
long-period constituents already named be represented by the equation 
y=A cos (at+a)+B cos (bt+ 8)+C cos (at+y) (405) 
+D cos (dt+6)+E cos (et+e) 
268. For convenience in this discussion let ¢ be reckoned from the 
11.5th solar hour of the first day of series instead of the midnight 
beginning that day. Every value of ¢ to which the daily means refer 
will then be either 0 or a multiple of 24. 
Let' A’, B’, O7, D’, and E’, equal 
A cos a, B cos B, C cos y, D cos 6, and F cos e«, respectively, and 
JACOBY CX DO tendon Tequal 
—A sin a, —Bsin 8, —C sin y, —D sin 6, and —F sin ¢, respectively. 
(406) 
Then formula (405) may be written 
y= A’ cos at+ B’ cos bt+C’ cos ct+D’ cos dt+-K’ cos et 
+A’ sin at+B” sin bt+C” sin ct+D” sin dt+E” sin et (407) 
In the above equation there are 10 unknown quantities, A’, A’’, 
B’, B’’, etc., for which values are sought in order to obtain from them 
the amplitudes and epochs of the five long-period constituents. The 
most probable values of these quantities may be found by the least 
square adjustment. 
269. Let 4, yo, . . - . Y3es represent the daily means for a 365 day 
series, as obtained from observations. If we let n be any day of the 
series, the value of ¢ to which that mean applies will be 24(n—1). 
By substituting in formula (407) the successive values of y and the 
values of ¢ to which they correspond, 365 observational equations are 
formed as follows: 
