I took mass readings of the three masses and the screw, and removed mass from one of the elongated masses, maintaining the mass distribution as much as possible:
The graph shows that the relationship between mass removed and time gained could be assumed to be linear, (for very small changes in mass). Therefore the data can be entered into a statistical calculator and linear regression performed:
M = bT + a
b = -0.312 gradient (M/T)
a = 9.34 intercept (where T=0)
so M = -0.312T + 9.34
Linear regression is the mathematical procedure used to find the "best fit" line through a set of points on a graph like the one shown.
If the correlation coefficient were small, you would try to determine from the graph whether the relationship were linear or non-linear (polynomial) or random. If it is not linear, obviously linear regression would not apply.
The correlation coefficient for the above data is: r = -0.9986, showing that the relationship between mass removed and change in time keeping could be assumed to be linear within the range measured in this experiment for the purpose of repair. The coefficient value is negative because the gradient is negative. A perfect correlation coefficient would be ±1.
To find the change in time per gram of mass removed, we need the reciprocal of the gradient because it reflects the mass to be removed per minute of time to be gained (and since the mass decreases as the time increases, the gradient is negative). So,
T/M = 1/b = 1/-0.312 = -3.2079 = -3 min. 12.5 sec. per gram per 24 hrs.
To find the mass we need to remove, we need to find where the time change (i.e. gain or loss) is zero, the intercept, which is where T = 0, so M = 9.34 grams.
Since my scale was accurate of 0.1 g., I decided to remove mass up to 9.2 grams (or 4.6 grams from each side).
A test run revealed that the clock gained 46 seconds in 24 hours, an error small enough to correct using the regulator to lengthen the suspension spring slightly.
The relationship between mass and time keeping could be assumed to be linear for small amounts of mass removed relative to the total mass of the pendulum, at least for the purpose of repairing Atmos clocks. Removing half a gram from each side will cause the Atmos clock to gain 3 min. 12.5 sec. per 24 hours.
A change in the strength of the suspension spring will change the gradient, i.e. the time gained per gram removed. Therefore, the 3:12.5 conclusion is a guideline because of possible variations in the manufacture of Atmos suspension springs. I would urge caution in removing the full calculated mass at once: for example, instead of removing 9.3 grams as calculated, try removing 7 or 8 grams and take a reading.
As more mass is removed, the angle of swing of the pendulum, or its amplitude of oscillation, increases. Removing too much mass aggravates this unintended side effect and could cause overbanking. Take note of the angle of swing during the initial test run. Another reason to remove 7 or 8 grams instead of the full calculated amount. If the first reading you take is not approximately 28 minutes slow per 24 hours, do not use this method.
Another reason why caution should be used is that the relationship could not be assumed to be linear for larger relative masses removed. The relationship would show a curve, (where the period is proportional to the square-root of the mass).
I have used this method no fewer than five times in the last five years with consistent results. It works well for me. However, if you do not understand the physics and the maths behind this experiment, you would probably experience difficulties and should not use this method: use at your own risk!