HARMONIC ANALYSIS AND PREDICTION OF TIDES 93 
=A’ cos 04-B" cos 0-> . 
ale ’’ sin 0+ B”’ sin O+ . 
Y2—A’ cos 24a+B’ cos 24b-+- . 
+ A’’ sin 24a+B” sin 246+ . (408) 
Yse5— A’ cos 24 364a-+B’ cos 24 3640+ . 
+A’’ sin 24364a+B” sin 2436464 . 
270. A normal equation is now formed for each unknown quantity 
by multiplying each observational equation by the coefficient of the 
unknown quantity in that equation and adding the results. Thus, 
for the unknown quantity A’, we have 
yi: cos O= A’ cos’ 0+B’ cos 0 cos 0+ - 
+A” sin 0 cos 0+ B” sin 0 cos ogee 
Yy2 cos 24a= A’ cos” 24a+B’ cos 246 cos yi 
aa sin 24a cos 24a+ B” sin 246 cos Made 
Y365 COS (24 X364a) =A’ cos” (24 364a) 
+B’ cos (243646) cos (24 364a)4+ - 
+A” sin (24X364a) cos (24364a) 
+B” sin (243646) cos (24X364a)+ -+-.-:. 
Summing 
= Yn cos 24(n—1)a=A’ py cos’ 24(n—1)a 
+A” a sin 24(n—1l)a cos 24(n—1)a 
+B’ baie: 24(n—1)b cos 24(n—1)a 
+B" 5)sin 24(n—1)b cos 24(n—1)a 
+" "Sy cos 24(n—1)e cos 24(n—1)a 
+ on 5) sin 24(n—1)e cos 24(n—1)a 
+)’ "00s 24(n—1)d cos 24(n—1)a 
+)” = sin 24(n—1)d cos 24(n—1)a 
+ EH’ ot cos 24(n—1)e cos 24(n—1)a 
+E" >) sin 24(n—1)e cos 24(n—1)a (410) 
which is the normal equation for the unknown quantity A’. 
271. In a similar manner we have for the normal equation for the 
quantity A’ 
