Playing around with equations for a moment:-
We all know the following...
M = TAS / SQRT(Gamma * R * T)
and EAS = EAS * SQRT (Sigma)
Thus M = EAS * SQRT (sigma / (gamma * R * T))
clearly gamma and R are constants, so we can take those out, giving...
M = k * EAS * (sigma / T)^½
So, accepting that as you climb you start at a EAS value (we'll assume it's the same as IAS which is a reasonably approximation) and switch to a Mach number, the changeover point comes when...
M(climb)=k*EAS(climb) * (sigma / T)^½
Since M(climb), EAS(climb) and k won't change, we get a changeover condition of...
(sigma/T)^½ = M(climb) / (k*CAS(climb))
Just making a quick check on this, at FL200 (the height selected arbritrarily) my ISA tables show for ISA the left hand of this equation is 0.0463. Using the ISA-15°C table however it comes out at 0.0493.
So it would appear that the changeover condition isn't always at the same altitude.
Having said that, last time I was asked to calculate a climb profile we simply assumed ISA and since no Vne / Mne limits were approached I don't think anybody troubled greatly about the innacuracies and small reductions in optimal climb performance - we did declare a fixed changeover flight level.
G