HARMONIC ANALYSIS AND PREDICTION OF TIDES 85 
B= —¢(B)=true initial phase of constituent B at beginning of 
series. 
a=speed of constituent A. 
b=speed of constituent B. 
246. Formula (369) may be written 
y=A cos a cos at+ z B cos {(b—a)t+ B} cos at 
—A sin asin at—z B sin {(6—a)t+ 8} sin at 
=[A cos a+ > B cos { (b—a)t+B}] cos at 
—[A sin a+ B sin {(b—a)t+8}] sin at (370) 
The mean values of the coefficients of cos at and sin at of formula 
(370) correspond to the coefficients C, and S, of formulas (295) and 
(296) which are obtained from the summations for constituent A. 
247. Let A’ and a’= the uneliminated amplitude and initial phase, 
respectively, of constituent A, as obtained directly from the analysis. 
The equation of the uneliminated constituent A tide may be written 
y=A’ cos (at+a’)=A’ cos a’ cos at—A’ sin a’ sinat (871) 
Comparing (370) and (371), it will be found that 
A’ cos a’=mean value of [A cos a+ = B cos {(b—a)t+6}] (372) 
A’ sin a’=mean value of [A sin a+ 2 Bsin {(b—a)t+8}] (373) 
248. Let r=length of series in mean solar hours. Then the mean 
value of 
B cos {(b—a)t+8} within the limits t=0 and t=7, is 
1s cos ((b—a)t+ pdt =a [sin {(b—a)r+6}—sin 8B] 
__180 sin 3(b—a) 
a 43(b—a)r 
“B cos {4(b—a)7+ B} (374) 
The mean value of B sin { (6—a)t+ 6} within the same limits is 
1's sin {(b—a)t+ B}dt=— Papyles {(b—a)r+ 8}—cos 6] 
= 180 Se COB sin (30—a)r+- 8} (375) 
Substituting (374) and (375) in (372) and (373), and for brevity 
etting 
180 sin 4(6—a) 
T 
we have 
A’ cos a’ =A cos a+ > F, cos {4(b6—a)t+ B} (377) 
A’ sin a’=Asin a+ F, sin {3(6—a)r+ 8B} (378) 
Transposing, 
A cos a=A’ cos a’—2 F, cos {4(6—a)7+ B} (379) 
Asin a=A’ sin a’— F, sin {4(b—a)7+ B} (380) 
Multiplying (379) and (380) by sin @’ and cos a’, respectively, 
