Let's go through a units example problem for Newton's law of universal gravitation from [1] with some updated modern values.

Show that has the value . If the unit for mass is the pound, the unit for length is the foot, the unit for time is the seconds and the unit for force is the poundal. One foot contains [2] and one pound is [2] and the Gravitational constant is [2].

For this problem Kellogg is referencing as a constant of proportionality in Newton's law of universal gravitation whose value depends solely on the units chosen.

The answer to this problem is a straight forward unit conversion. We start with the value of the Gravitaitonal constant in CGS units and convert to FPS units

Let's convert out the dyn first to fundamental units, 1 dyn has units

so for CGS units

Given the unit conversions in the problem statement, we can then easily convert to FPS units

Multiplying out and cancelling terms yeilds

Noting that a poundal is

Bibliography

1

Kellogg, Oliver Dimon. Foundations of potential theory. 1929. Berlin [u.a.]: Springer.

2

The NIST Reference on Constants, Units, and Uncertainty