158 U. S. COAST AND GEODETIC SURVEY 
Table 27. Critical logarithms for Form 245.—This table .us de- 
signed for quickly obtaining the natural numbers to three decimal 
places for column (3) of Form 245 from the logarithms of column (2). 
The logarithms are given for every change of 0.001 in the natural 
number. Each logarithm given in this table is derived from the 
natural number that is 0.0005 less than the tabular number to which 
it applies. Intermediate logarithms, therefore, apply to the same 
natural number as the preceding tabular logarithm. For example, 
logarithms less than 6.6990 apply to the natural number 0.000 and 
logarithms from 6.6990 to 7.1760 apply to the natural number 0.001, etc. 
Table 28. Constituent speed differences.—The constituent speeds as 
given in table 2 were used in the computation of this table. 
Table 29. Elimination factors—These tables provide for certain 
constant factors in formulas (389) and (390). Separate tables for 
each length of series and different values for each term of the formulas 
are required. The tabular values are arranged in groups of three, 
determined as follows: 
So St (fh 
First value=logarithm of oe sin 4 (6—a)r. 
4(b—a)r 
& omnaid int 
Second value=natural number oe always taken as 
positive. Ne 
Rechte le 
hid alent pe ale ae eee “ay aOStinine: 
sin 4(b—a) 
or 4(6—a)r+180, if 7 is negative. 
4(b—a)r 
Table 30. Products for Form 245.—This table is designed for ob- 
taining the products for columns (6) and (7) of Form 245. 
Table 31. For construction of primary stencils——This table gives 
the differences to be applied to the solar hours in order to obtain the 
constituent hours to which they most nearly coincide. Each differ- 
ence applies to several successive solar hours, but for brevity only 
the first solar hour of each group to which the difference applies is 
given in the table. 
An asterisk (*) indicates that the solar hour so marked is to be used 
twice or rejected according to whether the constituent speed is greater 
or less than 15p, when in the summation it is desired to assign a 
single solar hour to each successive constituent hour. For the 
usual summations in which each solar hour height is assigned to the 
nearest constituent hour no attention need be given to the asterisk. 
The table is computed by substituting successive integral values for 
d in formula (243) and reducing the resulting solar hour of series (shs) 
to the corresponding day and hour. The solar hour to be tabulated 
is the integral hour that immediately follows the value of (shs) of 
the formula. If the fractional part of (shs) exceeds 0.5, the tabular 
solar hour is marked by an asterisk (*). The successive values of d, 
although used positively in formula (243), are to be considered as 
negative in the application of the table when the speed of the con- 
stitutent is less than 15p. When the constituent speed is greater than 
15p, the difference is to be taken as positive. All tabular differences 
are brought within the limits +24 hours and —24 hours by rejecting 
multiples of +24 hours when necessary, and for convenience in use 
all differences are given in both positive and negative forms. 
