HARMONIC ANALYSIS!) AND PREDICTION OF TIDES 
69 
202. To obtain the value of any coefficient S, such as S,, multiply 
each equation of (282) by sina p wu. 
obtain 
a=(n—1) 
>S h.snapw =A 
a=0 
aS 
wie 
By (278), (278), and (281) the quantities 
a=(n—1) 
2 
a=o 
a=(n—1) 
and") Ss, 
a=0 
and p except when m and p are equal. 
Sm 
a=(n—1) 
Pas 
a=o0 
Cn 
sn ap u 
a=(n—1) : 
>> cosamusinapu 
a=0 
a=(n—1) ; 
>) sinamusinapu 
a=(n—1) 
Bs 
a=0 
a= 
2 
m and » is less than 5 = and by (276), the quantity 
a=0 
and 
S,= 
Therefore, Aas (291) reduces to the form 
a=(n—1) 
h, sina p u=4 nS) 
2 
x te GEO oD 
a=0 
Sum the resulting equations and 
(291) 
sin a p wu and 
cos am wu sin a p u are zero for all the values of m and p; 
sin @ m usin a p u becomes zero for all the values of m 
In this case the limit of / for 
(n—1) | 
sin? a p u 
a=0 
(292) 
(293) 
203. By substituting (285), (289), (290), and (293) in (257), the 
following equation of a curve, which will pass through the n given 
points, will be obtained 
© 
Il 
a 
B 
I 
ey 
ahs) 
) 
cos a w| cos 6 
9 
ll 
° 
~o 
ll 
ol), 3 
h, sin a w | sin 6 
p © 
Ml 
—-~ Oo 
B 
—1) 
h, cos 2 au|cos 20 
» © 
tol 
B (=) 
—1) 
Sip sin2au|sin 20 
—1) 
h, coska | 
oO 
=(n—1) ; 
ha sin law |sinJ@ 
a=0 
1 
*If n is even and k= > this fraction is — instead of 2. 
n 
(294) 
