53 



that the complete analysis of the M group of components at a station 

 includes the separation of six components. Computation programs 

 for this purpose are shown on standard forms of the United States 

 Coast and Geodetic Survey. The analysis of the S group of com- 

 ponents includes the separation of four components; and of the K 

 group 2. All other components require but a single analysis. 



97. Augmenting factors. — In the computations of the components 

 (except those of the S group) the tidal height at each component 

 hour is taken as the average of the observed heights at the nearest 

 mean solar hour (par. 83). When the computation is made from 

 stencils in the form ordinarily used (par. 88), these observed heights 

 are scattered quite uniformly over an interval extending from one- 

 half a component hour before to one-half a component hour after 

 the exact com.ponent hour. 



It is graphically apparent from 

 figure 27 that on a sinusoidal curve 

 the average of these heights is 

 always somewhat less, numeri- 

 cally, than the height at the 

 middle of the period, and that a 

 small systematic error is intro- 

 duced by using the average value. 

 This error is readily corrected, 

 since on a cosine curve the mid 

 height has a constant ratio to the mean height over an arc of given 

 length. This ratio is called the augmenting factor. 



98. To determine the augmenting factor for a component wdiose 

 equation is y=A cos (at-\-a), let r be the length of the componen thour. 

 The average value of y, between the limits of to—Yor and to-\-}iris then: 



Figure 27. 



,J.+ .. 



(1/r) °^'- A cos {at+a)dt 



= (^/ar)[sin (afo+/2a?*+«)— sin (a^o— /2«^+a)] 

 = {Ajar) [sin (afo+ «+ /sar) — sin {atQ-\-a— %ar)] 

 =2 {Alar) cos (a^o+«) sin )iar (58) 



The ratio of the mid value to the mean value is then: 



A cos {atQ-\-a)l{2Alar) cos {at^-^-a) sin }2ar = ar/2 sin jiar (59) 



in which ar is an angle expressed in radians, whose equivalent, in 

 degrees, is xar/180°. The expression for the augmenting factor is 

 therefore: 



7r(ir/360° sin jiar 



