HARMONIC ANIALYSIS AND PREDICTION OF TIDES 161 
one tooth forward. The period of rotation of each shaft and gear is 
relative and refers to the time as indicated on the face of the machine, 
which for convenience is called dial time. 
Table 38. Coast and Geodetic Survey tide-predicting machine No. 2— 
Constituent gears.—This table contains the details of the gearing from 
the main vertical shafts to the individual constituent cranks. Column 
I gives the number of teeth in the bevel gear on the main vertical 
shaft; column II, the number of teeth in the gear on the intermediate 
shaft that meshes with the gear on the vertical shaft; column III, the 
number of teeth in the gear on the intermediate shaft that meshes with 
the gear on the constituent crank shaft; and column IV, the number 
of teeth in the gear on the crank shaft. 
For the long-period constituents the worm gear is taken as the 
equivalent of one tooth. For each of these constituents there is a 
short secondary shaft on which sliding gears are mounted, but the 
extra gears do not affect the speed of any of the crank shafts except 
that for constituent Sa in which case a ratio of 1:2 is introduced. 
The crank-shaft speed per dial hour for each constituent is equal to 
3qos¢ column I << ea nua gag 
column II’*column IV 
both values appearing in each of the columns II and III is to be taken 
as the value for the column. The column of ‘Gear speed per dial 
hour” contains the speeds as computed by the above formula. 
For comparison the table contains also the theoretical speed of each 
of the constituents and the accumulated error per year due to the 
difference between the theoretical and the gear speeds. 
For convenience of reference the table includes also the maximum 
amplitude settings of the constituent cranks. 
Table 39. Synodic periods of constituents.—This table is derived 
from table 28, the period represented by 360° being divided by the 
speed difference and the results reduced to days. 
Table 40. Day of year corresponding to any date.—This table is 
convenient for obtaining the difference between any two dates and 
also in finding the middle of any series. 
Table 41. Values of h in formula h=(1+7r°+2r cos x)?.—This table 
may be used with formulas (472) and (473) on page 149 to obtain 
constituent amplitudes for the prediction of hydraulic currents. 
Table 42. Values of k in formula kta Tog ge rhs table 
may be used with formulas (474) and (475) on pages 149-150 to 
obtain constituent epochs for the prediction of hydraulic currents. 
For constituent Sa the product of 
