HARMONIC ANALYSIS’ AND PREDICTION OF TIDES 57 
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 constituents with commensurable periods 
will have the same tabular values, and no distinction is made in the 
construction of the stencils. Thus, stencils for constituent M serve 
not only for M, but also for M3, My, Mg, ete. 
172. For the construction of a set of stencils for any constituent a 
preliminary set of the hourly height forms is prepared with days of 
series numbered consecutively beginning with 1 and each hourly 
height space numbered with its constituent hour as derived by the 
differences in table 31. The even and odd constituent hours are then 
transferred to separate sets of forms and the marked spaces cut out. 
In the Coast and Geodetic Survey this is done by a small machine 
with a punch operated by a hand lever. Spaces corresponding to the 
same constituent hour are connected by ruled lines which are num- 
bered the same as the hours represented. Black ruling with red 
numbering is recommended, the red emphasizing the distinction 
between these numbers and the tabulated hourly heights which are 
to be summed. 
173. When in use the stencils are placed one at a time on the sheets 
of tabulated heights, with days of series on stencils matching those on 
the tabulations, and all heights on the page corresponding to each 
constituent hour are then summed separately. For constituent S no 
stencils are necessary as the constituent hours in this case are identical 
with the solar hours. For constituents K, P, R, and T with speeds 
differing little from that of S, the lines joining the hourly spaces 
frequently become horizontal and the marginal sum previously ob- 
tained for constituent S becomes immediately available for the sum- 
mation at hand. In these cases a hole in the margin of the stencil 
for the sum replaces the holes for the individual heights covered by 
the sum. 
SECONDARY STENCILS 
174. After the sums for certain principal constituents have been ob- 
tained by the stencils described in the preceding section, which for con- 
venience will be called the primary stencils, the summations for 
other constituents may be abbreviated by the use of secondary sten- 
cils which are designed to regroup the hourly page sums already ob- 
tained for one constituent into new combinations conforming to the 
periods of other constituents. Certain irregularities are introduced 
by the process, but in a long series, such as 369 days, these are for the 
most part eliminated, and the resulting values for the harmonic con- 
stants compare favorably with those obtained by use of the primary 
stencils directly, the differences in the results obtained by the two 
methods being negligible. For short series the irregularities are less 
likely to be eliminated, and since the labor of summing for such a 
series is relatively small, the abbreviated form of summing is not 
recommended. As the length of series increases the saving in labor 
by the use of the secondary stencils increases, while the irregularities 
due to the short process tend to disappear. It is believed that the 
use of the secondary stencils will be found advantageous for all series 
more than 6 months in length. 
