**Straw-Man**

Proposed Mathematics of Atomic Time (UTC) measurement comparison calculations:

Assuming our clocks run at a uniform rate (some straight line of any slope) then:

Measure "r" as the difference between of Atomic time and
individual Clock time = r_{1}, r_{2}, r_{3}….
r_{N.} at time period N.
Assumes clocks are not set back to atomic time at the beginning of each
time period. (+ = Clock reading faster than Atomic Time. - = Clock reading slower than Atomic
time)

First derivative or slope in (sec/day) is then R_{N}
= r_{1}/d_{1}, (r_{2} -
r_{1})/d_{2},
(r_{3} - r_{2})/d_{3, } . . . Where d_{N }= Number of
days (to at least several decimal places) between each measurement N.

Second derivative or acceleration of time
(sec/day^{2}) is then:

C_{N} =
(R_{2 }- R_{1})/d_{2}, (R_{3 }-
R_{2})/d_{3}, . . . (R_{N} -
R_{N-1})/d_{N})

Next select the most reliable clocks using the resulting
standard deviation as a reference. Average all Second derivative C_{N}
readings for selected clocks during each time period or AC_{N} = (C_{2} +
C_{3} + C_{4}….+ C_{N})/(N-1) Where C_{N} is second
derivative for each of N-1 clocks.

First Integral of average AC_{N} or FI_{N }=
AC_{1}*d_{1,
}(AC_{1}*d_{1}+AC_{2}*d_{2})_{,
}(AC_{1}*d_{1}+AC_{2}*d_{2}+
AC_{3}*d_{3})_{, }. . .

Integral of
FI_{N } or
SI_{N}= FI_{1}*d_{1,
}(FI_{1}*d_{1}+FI_{2}*d_{2})_{,
}(FI_{1}*d_{1}+FI_{2}*d_{2}+
FI_{3}*d_{3})_{, }. . .

This second integral then becomes the filtered constructed curve of the average comparison to atomic time.

Note that all straight-line trends of any constant slope in the original data are filtered out, as desired. Only the changing situation is plotted as the results.