HAEMOI^IC ANALYSIS AISTD PEEDICTIOlSr OF TIDES. 105 



larly, the disturbing effects of other short-period components may be 

 represented by 



1 „ sin 12h ,. . 



etc. 



The combined disturbing effect of all the short-period components 

 may, therefore, be represented by the equation 



1 , sin 12a . ^ , s 

 y = 'ya + yb + etc. =24^ sin ^a ^^^ (ai + o') 



+ ^B '^^ cos {U + /3) + etc. (422) 



This formula is adapted to use on the tide-computing machine. 

 With the component cranks set in accordance with the coefficients 

 and initial epochs of the above formula, the machine will indicate 

 the values of y corresponding to successive values of t. The values of 

 y desired for the clearances are those which correspond to t at the 11.5 

 hour on each day. Thus, the clearance for each successive day of 

 series may be read directly from the dials of the machine. In practice, 

 it may be found more convenient to use the daily sums rather than 

 the daily means for the analysis. In this case the coefficients of the 

 terms of (422) should be multiplied by the factor 24 before being 

 used in the tide-computing machine. 



Assuming that all the daily sums are used in the analysis, the 

 augmenting factor represented by formula (329) which is used for 

 the short-period component is also applicable to the long-period com- 

 ponents, with p representing the number of component periods in a 

 component month or a component year. Thus, for components Mm 

 and Sa, p equals 1, and for Mf , MSf, and Ssa, p equals 2. For the long- 

 period components a further correction or augmenting factor is 

 necessary, because the mean or sum of the 24 hourly heights of the day 

 is used to represent the single ordinate at the 11.5 hour of the day. 



If we let formula (416) be the equation of the long-period component 

 sought, formula (420) mil give the mean value of the 24 ordinates of 

 the day which, in the grouping for the analysis, is taken as represent- 

 ing the 11.5 hour of the day or the ^d hour of the series. Since the true 

 component ordinate for this hour should be A cos {at^ + a), it is 



Qi rj X/"/ 



evident that an augmenting factor of 24-^ — :~y- must be applied to the 



Sill L^Q/ 



mean ordinates as derived from the sum of the 24 hourly heights of the 

 day in order to reduce the means to the 11.5 hour of each day. 



The complete augmenting factor for the long-period components 

 will therefore be obtained by combining the above with (329) to 

 obtain 



-P xgj^ (423) 



„^ . 15© sm 12a 

 24 sm —^ 



Values obtained from formula (423) are given in Table 20. 

 The following method of reducing the long-period tides, which 

 conforms to the system outlined by Sir George H. Darwin, differs 



