To add a couple of minor comments to Bellerophon's observations, it is usual to include the transducer loss (usually called something like the 'recovery factor') to arrive at
TAT = SAT(1 + kM^2/5)
So far as the alternative TAS equation is concerned, one should be aware that the coefficients are fairly sensitive to the constant values used in their derivation, so you might find some variation if you do the sums yourself. For aircraft where temperature rise is a problem which needs to be considered, you normally have a mach reading so there is little obvious point in playing with TAS.
Contemplating acegreaser's question, one needs to dig out the old physics textbook (dust exits stage left)...
Speed of sound in a medium (air, steel, whisky, ... whatever) is normally described in terms of
speed of sound = square root(bulk modulus/density)
where the bulk modulus (of elasticity) is a fairly simple measure of medium compressibility. Thus it is not just a matter of considering the medium density by itself. If the terms in the above equation are massaged a trifle, things substituted and the like, the equation ends up something like the more useful ratio expression
a1/a2 = square root (t1/t2)
where a is the speed of sound for two temperature states and t is the measure of the temperature state in absolute units (either Kelvin or Rankin, depending on your preference for centigrade, sorry, celcius, or farenheit)
Thus it follows that the speed of sound increases with increased temperature which is what we see - speed of sound is higher near sea level (higher OAT) and lower at cruise level (lower OAT)
[ 20 October 2001: Message edited by: john_tullamarine ]