HAEMOlSriC AISTAX-YSIS AjSTD PEEDICTION OF TIDES. 



89 



This augmenting factor may be %vritten 



rhp 



24a sin 



15hp 

 2a 



irp^ 



24 sin 



15pi 



7 sin 



ISOhp 



. 12Q0hp 

 sm 



(336) 



The first factor of the above is to be omitted if the primary sum- 

 mations are for component S. It will be noted that the middle 

 factor is the same as the augmenting factor that would be used if 

 component B had been subjected to the primary summations. 



28. REDUCTION OF EPOCHS AND AMPLITUDES TO MEAN VALUES. 



In equation (321), page 84, 



y = A cos (p 9-^) 



the quantity ( — f ) is the phase of the component at the time 6 equals 

 zero — that is, at the beginning of the series — and p 6 is equal to f at 

 the time y is a maximum; that is, at the time of the component high 

 water. The value of f will therefore depend upon the time of the 

 beginning of the series, which is more or less arbitrary, and the f 's of 

 any component determined from different series of observations are 

 not directly comparable. Expressions for the theoretical phases, or 

 arguments, of the principal lunar components are represented in 

 formula (100), page 35, and a general expression for this argument as 

 modified by a constant k for a particular locality is given by formula 

 (101), page 39. The last formula is an equivalent of the angle of 

 (321) and may be written 



{at+Vo+u-K)=p d-^ (337) 



In the above the variable angle p 6 = at, the angle d having its zero 

 value at the beginning of the series when t equals zero. 

 Then 



or (338) 



'c=r+ Vo+u 



The significance of the expression (Vo + u) was discussed in 

 section 10. 



In (338) K is a constant that is independent of the beginning of the 

 series, and it is called the opoch of the component. The k's as deter- 

 mined from different series of observations for the same locality are 

 comparable. 



The angle k may be graphically represented by Figures 1 and 11. 

 In Figure 1, we have a simple representation of a single component. 

 In this figure changes in the phase or angle are measured along the 

 horizontal line, positive change toward the right and negative change 

 toward the left. The full vertical line indicates the beginning of the 

 series, at which time the angle p 6, or at, equals 0. At the left of this 

 vertical line, the symbol of a moon (M) indicates the zero value of the 

 equilibrium argument that precedes the beginning of the series. 

 For the principal lunar or solar component, this will be simultaneous 

 with a transit of the mean moon (modified by longitude of moon's 



