HARMONIC ANALYSIS! AND PREDICTION OF TIDES 55 
it is desired to obtain the integral constituent hour corresponding 
most nearly with each solar hour, the (chs) should be taken to the 
nearest integer by rejecting a fraction less than 0.5, or counting as 
an extra hour a fraction greater than 0.5, or adopting the usual rule 
for computations if the fraction is exactly 0.5. The constituent 
hour of the constituent day (ch) required for the construction of the 
stencils may be obtained by rejecting multiples of 24 from the (chs). 
165. In the application of the above formula it will be found that the 
integral constituent hour will differ from the corresponding solar 
hour by a constant for a succession of solar hours, and then, with the 
difference changed by one, it will continue as a constant for an- 
other group of solar hours, etc. This fact is an aid in the prepara- 
tion of a table of constituent hours corresponding to the solar hours 
ef the series, as it renders it unnecessary to make an independent 
calculation for each hour. Instead of using the above formuta for 
each value the time when the difference between the solar and con- 
stituent hours changes may be determined. The application of the 
differences to the solar hours will then give the desired constituent 
hours. 
166. Formula (241) is true for any value of (shs), whether integral or 
fractional. It represents the constituent time of any instant in the 
series of observations in terms of the solar time of that same instant, 
both kinds of time being reckoned from the beginning of the series 
as the zero hour. The difference between the constituent and the 
solar time of any instant may therefore be expressed by the following 
formula: 
a~ldp 
15p 
167. If the constituent day is shorter than the solar day, the speed a 
will be greater than 15p, and the constituent hour as reckoned from 
the beginning of the series will be greater than the solar hour of the 
same instant. If the constituent day is longer than the solar day 
the constituent hour at any instant will be less than the solar hour 
of the same instant. At the beginning of the series the difference 
between the constituent and solar time will be zero, but the difference 
will increase uniformly with the time of the series. As long as the 
difference does not exceed 0.5 of an hour the integral constituent 
hours will be designated by the same ordinals as the integral solar 
hours with which they most nearly coincide. Differences between 
0.5 and 1.5 will be represented by the integer 1, differences between 
1.5 and 2.5 by the integer 2, etc. If we let d represent the integral 
difference, the time when the difference changes from (d—1) to d, 
will be the time when the difference derived from formula (242) 
equals (d—0.5). Substituting this in the formula, we may obtain 
15p 
a~15p 
Difference=75- (shs) ~ (shs) = (shs) (242) 
(shs) = (d—0.5) (243) 
in which (shs) represents the solar time when the integral difference 
between the constituent and solar time will change by one hour from 
(d—1) tod. By substituting successively the integers 1, 2, 3, etc., 
for d in the formula (243) the time of each change throughout the 
series may be obtained. The value of (shs) thus obtained will 
