Just came upon this thread while having an idle browse.
I'm sure there'll be a reason for the subtle difference
Bob's given the basic answer. However, as this question arises regularly, it probably is worth knowing a bit of the history in order to understand what's what.
Traditionally, the TAS calculation was figured using incompressible equations which work fine for low performance aircraft pottering along at less than, say, Mach 0.2 to 0.3 or so. For this style of calculation, setting Hp against OAT in the TAS cutout sets Hd in the second cutout or, more specifically, sets 1/√σ (where σ is the density ratio - local density divided by ISA SL density) on the outer CD scale (the outside scale on the calculation side of the CR computer) against the "10" index on the inner scale.
This then sets you up for a simple multiplication on the CD scales in the usual manner of the slide rule to convert CAS to TAS.
For example, if we were at 10,000 ft, then we could look up the value of σ in a suitable reference, say
onlinelibrary.wiley.com/doi/pdf/10.1002/9781118534786.app1, which is about 0.7385, take the square root, which is about 0.86 and then take the reciprocal (ie 1 divided by the number) and end up with about 1.16. If you set Hd 10,000 ft in the TAS cutout window and check the outer scales, you will see "10" on the inner scale nicely aligned with (or pretty close to - depending on the engraving accuracy of your particular CR or Dalton calculator) 1.16 on the outer scale. Say you were motoring along at 130 KCAS, then, if we go around to 130 on the inner scale, we can read off a TAS of around 151-152 KTAS on the outer scale. That's all the TAS cutout calculation does, regardless of whether you are using a Dalton or CR original or clone.
Now, if we go back to WW2, the two principal military calculators were the Dalton (developed by Dalton and used by the Allies) - which was known by the original USAAF designation of E-6B - and the DR2 (developed by Knemeyer and used by the Axis). The DR2 was quite a neat bit of kit and did the nav triangle calculation by trigonometry. Very elegant solution and super easy to use. For whatever reason, post war, the DR2 largely fell into disuse.
However, Ray Lahr (a UAL pilot) developed a nav computer in the 1950s which traced its roots back to the DR2, although the nav solution was a little less elegant in that it combined a graphical and trigonometric solution. Co-incidentally, another American (Franz Huber) had developed a TAS computer in the late 1940s, based on compressible flow equations, and Lahr co-opted Huber's device into his calculator. Lahr and UAL sold the rights to Jeppesen and the rest is history. The CR has both the compressible and incompressible solutions incorporated into the device - you take your pick as to which you use.
The extremely large numbers of aircrew during the war who were exposed to one or the other of the Dalton or DR2 led to all the other styles of nav computer falling by the wayside and, with Jeppesen's marketing power, we now have, essentially, the choice of Dalton or CR or clones for pilot use.
The OP's posts relate to the use of the Huber method as incorporated into the CR nav computer.
In this solution, we set CAS against Hp, which gives Mach Number (in the same way the Machmeter calculates M). You will see a small Mach cutout which logs the specific M value for any given combination of CAS/Hp. So, for instance, if we combine 240 KCAS and 20,000 ft Hp you get a MN of around 0.526. From MN, we can calculate temperature rise and Huber's rather clever setup allows us to overlay CT values to read off both temperature rise and TAS. The setout for the Huber method is a series of radial plots of the equations and gets a bit messy to explain. If you are interested in playing with the technique, you might like to download Huber's patent and work your way through his explanation.
You will see that, for low MN, the two solutions are, in essence, the same. For the OP's example of concern, I get (using an ASA CR clone) around 192 KTAS using the incompressible solution (as per Bob's post) and, for the compressible solution (as per the OP's post) around 190 KTAS. However, note that the MN is around 0.3 so we are getting into the region where we would expect to see some divergence between the two solutions. Had I taken the trouble to pull out my Dalton computer, I would have obtained around 192 KTAS as the calculation on the two computers is the same.
We see an error creeping into the incompressible solution, which sees the KCAS values increasingly overread as the MN increases towards Mach 1. The reason for this is that the instrument makers couldn't come up with an ASI construction which accounted for all the story so they settled for a design which was OK for SL standard conditions. If we want to use the incompressible solution, we need to account for the compressible flow errors inherent in the KCAS readings and, to do this, we resort to a trick called KEAS. The following link gives a graph of the error (and I suspect that graph has been taken from Hurt's book "Aerodynamics for Naval Aviators").
www.bing.com/images/search?view=detailV2...jaxhist=0&ajaxserp=0
Most of the Dalton clones don't give you a convenient means of running the correction. A few do but Huber's solution is much more straightforward and elegant.
Getting back to the OP's concern, the difference is subtle at low Mach numbers but gets pretty noticeable in the transonic range.
