56 “U. S. COAST AND GEODETIC SURVEY 
generally be a mixed number; that is to say, the times of the changes 
will usually come between integral solar hours. The first integral 
solar hour after the change will be the one to which the new difference 
will apply if the usual system of distribution is to be adopted. In 
this case we are not concerned with the exact value of the fractional 
part of (shs) but need note only the integral hours between which 
this value falls. 
168. If, however, the second system of distribution should be 
desired, it should be noted whether the fractional part of (shs) is 
greater or less than 0.5 hour. With a constituent day shorter than 
the solar day and the differences of formula (242) increasing positively, 
the application of the differences to the consecutive solar hours will 
result in the jumping or omission of a constituent hour at each change 
of difference. Under the second system of distribution each of the 
hours 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 constituent 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 (shs) in (248) is 
less than 0.5 hour, the old and new differences will both be applied to 
the preceding integral solar hour; but if the fraction is greater than 0.5 
hour the old and new differences will be applied to the integral solar 
hour following the change. 
169. With a constituent day longer than the solar day and the differ- 
ences of formula (242) increasing negatively, the application of the 
differences to the consecutive solar hours will result in two solar 
hours being assigned to the same constituent 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 (shs), 
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. 
170. Table 31, computed from formula (243), 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 constituent 
hour of the constituent day rather than with the constituent hour 
of the series, and these differences may be applied 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 applied, and at other 
times using the positive difference, the necessity for adding or 
rejecting multiples of 24 hours from the results is avoided. 
171. The tabulated solar hour is the integer hour that immediately 
follows the value for the (shs) is formula (243). An asterisk (*) 
indicates that the fractional part of the (shs) 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 will be used with 
both old and new difference to represent two constituent hours, or 
will be rejected altogether according to whether the constituent day 
