Just an idea but adding a capital to a running system, looks like to start a new system from zero.
Following your example is like you start a TS 9 years ago with 500 and another one one year ago. So the square root, and the other relevant parameters, may be calculated just adding the values of the two systems.
I have not the capabilities to prove it mathematically but it make sense to me: if the square root rule is correct for 2 user starting 2 TS independently one 9 years ago and one last year, should be correct for the sum of them...

ciao