3. The complete calculationThe method followed in the sample problem can be extended to the complete calculation. Given the tidal record H(t) over a suffuciently long time interval [0,P],
How were these calculations made? In terms of calculus, the average value of H(t)sin(v_{k}t) over the interval [0,P] is (1/P) times the integral from 0 to P of H(t)sin(v_{k}t)dt. With a modern computer a computation like this is not a problem, once H(t) has been presented as a file the computer can read. In the nineteenth and early twentieth century other methods were needed. One method, partially explained in Schureman, is the use of stencils overlaid on the hourly records for H(t) (regularly tabulated in columns) so that the terms with a common trigonometric factor would appear in a linked series of windows. The amount of calculation is still substantial, and must be completely repeated for each port. Kelvin's Harmonic Analyser. Kelvin devised a mechanism for implementing this calculation.
This machine is designed to compute the eleven coefficients A_{0}, A_{1}, B_{1}, ... A_{5}, B_{5}. The tidal record is diplayed on the horizontal drum in the center of the picture. The drum is linked by gearing to the eleven disks, part of eleven diskspherecylinder integrators. If the drum advances t units, the leftmost disk rotates t radians, the next two disks rotate sin (v_{1}t)/v_{1} and cos(v_{1}t)v_{1} radians, respectively, the third and fourth disks rotate sin (v_{2}t)/v_{2} and cos(v_{2}t)/

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