HARMONIC ANALYSIS!) AND PREDICTION OF TIDES 53 
periods according to whether the constituent is diurnal, semidiurnal, 
etc. The constituent day is divided into 24 equal parts, the beginning 
of each part being numbered consecutively from 0 to 23 and these are 
known as constituent hours. 
157. To carry out strictly the plan described above would require 
separate tabulations of the heights of the tide at different intervals 
for all constituents of incommensurable periods, a procedure involving 
an enormous amount of work. In actual practice the tabulated solar 
hourly heights are used for all of the summations, these heights being 
assigned to the nearest constituent hour. Corrections are afterwards 
applied to take account of any systematic error in this approximation. 
158. There are two systems for the distribution and assignment of 
the solar hourly heights which differ slightly in detail. In the system 
ordinarily used and which is sometimes called the standard system, 
each solar hourly height is used once, and once only, by being assigned 
to its nearest constituent hour. By this system some constituent 
hours will be assigned two consecutive solar hourly heights or receive 
no assignment according to whether the constituent day is longer or 
shorter than the solar day. In the other system of distribution, each 
constituent hour receives one and only one solar hourly height neces- 
sitating the occasional rejection or double assignment of a solar hourly 
height. The difference in the results obtained from the two systems 
is practically negligible but the first system is generally used as it 
affords a quick method of checking the summations. 
STENCILS 
159. The distribution of the tabulated solar hourly heights of the 
tide for the purpose of the harmonic analysis is conveniently accom- 
plished by a system of stencils (fig. 10) which were devised by L. P. 
Shidy of the Coast and Geodetic Survey early in 1885 (Report of 
U.S. Coast and Geodetic Survey, 1893, vol. I, p. 108). Although 
the original construction of the stencils involves considerable work, 
they are serviceable for many years and have resulted in a very great 
saving of labor. These stencils are cut from the same forms which 
are used for the tabulation of the hourly heights of the tide and 106 
sheets are required for the summation of a 369-day series of observa- 
tions for a single constituent. Separate sets are provided for different 
constituents. Constituents with commensurable periods are included 
in a single summation and no stencils are required for constituents, 
eh So, on etc. 
160. The use of the stencils makes a standardized form for the 
tabulation of the hourly heights essential. This form (fig. 9) is a 
sheet 8 by 10/4 inches, with spaces arranged for the tabulation of the 
24 hourly heichte of each day in a vertical column, with 7 days of 
record on each page. The hours of the day are numbered consecu- 
tively from 0° at midnight to 23" at 11 p.m. When the tabulated 
heights are entered, each day is indicated by its calendar date and 
also by a serial number commencing with 1 as the first day of series. 
The days on the stencil sheets are numbered serially to correspond 
with the tabulation sheets and may be used for any series regardless 
of the calendar dates. 
161. The openings in the stencils are numbered to indicate the 
constituent hours that correspond most closely with the times of the 
