HAKMOK^IC AiSTALYSIS AND PREDICTION OF TIDES. 69 



part of (sTis) but need note only the integral hours between which 

 this value falls. 



If, however, the second system of distribution should be desired, 

 it should be noted whether the fractional part of (sTis) is greater or 

 less than 0.5 hour. With a component day shorter than the solar 

 day and the differences of formula (263) increasing positively, the 

 application of the differences to the consecutive solar hours will 

 result in the jumping or omission of a component hour at each 

 change of difference. Under the second system of distribution each 

 component must be represented, and it will therefore be necessary 

 in this case to apply two consecutive differences to the same solar 

 hour to represent two consecutive component hours. The solar hour 

 selected for this double use will be the one occurring nearest to the 

 time of change of differences. If the fractional part of the (sJis) in 

 (264) is less than 0.5 hour, the old and new differences wiU both be 

 applied to the preceding integral solar hour; but if the fraction is 

 greater than 0.5 hour the old and new difference will be applied to 

 the integral solar hour following the change. 



With a component day longer than the solar day and the differ- 

 ences of formula (263) increasing negatively, the apphcation of the 

 differences to the consecutive solar hours will result in two solar 

 hours being assigned to the same component hour at each change of 

 differences. Under the second system of distribution this must be 

 avoided by the rejection of one of the solar hours. In this case the 

 integral solar hour nearest the time of change will be rejected, since 

 at the time of change the difference between the integral and the true 

 difference is a maximum. Thus, if the fractional part of the (sJis), 

 is less than 0.5 hour, the preceding solar hour will be rejected; but 

 if the fraction is greater than 0.5 hour the next following solar hour 

 will be rejected. 



Table 31, computed from formula (264), gives the first solar hour 

 of the group to which each difference applies when the usual system 

 of distribution is adopted. Multiples of 24 have been rejected from 

 the differences, since we are concerned only with the component 

 hour of the component day rather than with the component hour 

 of the series, and these differences may be apphed directly to the solar 

 hours of the day. For convenience equivalent positive and negative 

 differences are given. By using the negative difference when it does 

 not exceed the solar hour to which it is to be apphed, and at other 

 times using the positive difference, the necessity for adding or 

 rejecting multiples of 24 hours from the results is avoided. 



The tabulated solar hour is the integer hour that immediately 

 foUows the value for the (slis) in formula (264). An asterisk (*) 

 indicates that the fractional part of the (sJis) exceeds 0.5, and that 

 the tabular hour is therefore the one nearest the exact value of (shs) . 

 If the second system for the distribution of the hourly heights is 

 adopted, the solar hours marked with the asterisk wlQ be used with 

 both old and new difference to represent two component hours, or 

 wiU be rejected altogether according to whether the component day 

 is shorter or longer than the solar day. If the tabular hour is un- 

 marked, the same rule of double use or rejection will apply to the 

 untabulated solar hour immediately preceding the tabular unmarked 

 hour. For the ordinary stencils no attention need be given to the 

 asterisks. By the formula components with commensurable periods 



