how gold gets 4-104

How does Gold get 4-105? PhL 7.24.12 [ Here I realize that Goldstein is using one of the "cross velocities" in his discussion, and this is now all explained in frames doc. ] First, how does he get (4-104) ? In my notation, here is what he does (dr'/dt)S = (dr'/dt)S' + ω x r' or v'S = v'S' + ω x r' which with the S↔S' rule would translate into (dr/dt)s = (dr/dt)r + ω x r or vs = vr + ω x r Therefore we see that he is using vs = v'S vr = v'S ' Therefore, we find that these are the translation rules (including S↔S') r' → r v'S → vs v'S' → vr a'S → as a'S'→ ar ______________________________________________ Start with (4-104) vs = vr + ω x r r = terrestrial Apply the G rule to vs (dvs/dt)s = (dvs/dt)r + ω x vs // agrees with first line of 4-105 Now apply (d/dt)r to 4-104 (dvs/dt)r = (dvr/dt)r + ω x (dr/dt)r = ar + ω x vr ar = (dvr/dt)r Install this into the previous line to get (dvs/dt)s = ar + ω x vr + ω x vs Now use 4-104 in here to get (dvs/dt)s = ar + ω x vr + ω x [vr + ω x r] as = (dvs/dt)s as = ar + 2 ω x vr + ω x (ω x r) // agrees with second line of 4-105 **************************************************************************** How do I translate this all to my notation? I will try my initial idea 4-104 says for me vr → (dr'/dt)S' = v'S' vs → (dr'/dt)S = v'S The if this is correct, the translation of (4-104) is this v'S = v'S' + ω x r' r' = terrestrial = frame S' This is one of my supposedly valid equations which leads me to the idea shown for vr and vs. Now apply the G rule to v'S = vs (dv'S/dt)S = (dv'S/dt)S' + ω x v'S // my version of first line of 4-105 Now apply (d/dt)S' to 4-104 assuming ω = constant v'S = v'S' + ω x r' (dv'S/dt)S' = (dv'S'/dt)S' + ω x (dr'/dt)S' = a'S' + ω x v'S' Install this into two lines back to get (dv'S/dt)S = a'S' + ω x v'S' + ω x v'S Now in the last term use v'S = v'S' + ω x r' and we then get (dv'S/dt)S = a'S' + ω x v'S' + ω x v'S' + ω x (ω x r') which says a'S = a'S' + 2 ω x v'S' + ω x (ω x r') and all is well. ******************************************* The only problem is that I think I have showed in Gold Rule version 8 that a'S = a'S' + ω x v'S (7.2) and then this becomes, using 6 lines above, a'S = a'S' + ω x v'S' + ω x (ω x r') and this just does not look the same! as 3 lines above!!! I am then missing the factor of 2 !