51 



and T; (2IT) y cos at dt is twice the mean value of y cos af 



J 

 between the same hmits ; and that similarly all of the remaining values 

 of the coefficients are twice the mean values of the expressions inte- 

 grated. If then the values of y, or the determined heights of the 

 component at its successive component hours, are designated as ho, 

 hi, A2 ■■■ ^23, it immediately follows that: 



Ho=y2,(ho + k + h2-{- ... +/i23) (53) 



The angle at has the value of —15° at the end of the first com- 



24 



ponent hour, 30° at the end of the second component hour, etc. 



Twice the average values of y cos at; y sin at; y cos 2at; y sin 2at 



are then: 



Ci = }i2(ho cos + Ai cos 15° + /i2 cos 30° + . + A23 cos 345°) (54) 

 Si^Yniho sin O + Ai sin 15° + /i2 sin 30° + .+A23 sin 345°) (55) 

 C2=}U{ho cos + Ai cos 30° + /i2 cos 60° + .+A23 cos 330°) (56) 

 S2=yi2(ho sin + Ai sin 30°+A,2 sin 60° + . + /^23 sin 330°) (57) 

 and so on. 



Equation (53) merely expresses the evident fact that the elevation 

 of mean sea level is the mean of the heights at the component hours. 

 The amplitude and initial phase of the diurnal component are de- 

 termined by computing C] and Si from equations (54) and (55) and 

 applying to them the relations expressed by equations (47) and (48). 

 The amplitude and initial phase of the semidiurnal component are 

 similarly derived from the computed values of Co and ^2; and those of 

 other components from the corresponding coefficients, the equations 

 for determining which may be written by analogy to equations (54) 

 to (57). 



The computations of the coefficients from equations (54) to (57) 

 may be greatly abbreviated by combining the terms whose sine or 

 cosine factors have the same numerical value. For example, in find- 

 ing the values of C2 and S2 from equations (56) and (57), it is apparent 

 that these factors for A, 2 are respectively cos 360° and sin 360°, those 

 for ^13 are cos (360° + 30°) and sin (360° + 30°) and so on. The suc- 

 cessive factors for the last 12 terms are consequently the same as for 

 the first 12 terms. Furthermore, the factors for h^ are cos 180° and 

 sin 180°, respectively. Since cos (180 + 0) = — cos0 and sin (180°+</)) 

 = — sin 0, the successive factors for the second 6 terms are equal but 

 opposite in sign to those of the first 6 terms. 



95. Example. — Taking the heights of the M group of components 

 at their component hours at Sitka, computed in paragraph 86, the 

 computation of the amplitude and initial phase of the AI2 component 



