Another problem I find with this experiment is the expectation itself. I decided to study the effect on a very simplistic basis to undersand the nature of this second order effect. While my calculation method is longer than the formula, it is simpler and retains the understandings of the inner workings of the effect instead of relying on faith in the correctness of a formula. The purpose of my site is to communicate close to the level of laymen so that anyone examining it can remain confident in the correctness of what is being conveyed.
 
 
The reason the reading is so very small in this experiment is because of the fact that the light does not simply go with or against the flow but it does both because of the reflection. Many people erroneously assume that because the light is sped up in one direction but then equally slowed down on the return trip that there is an exact cancellation. This is not true of course and I've made some simple diagrams to help understand the concept of this second order effect.


Imagine a swimmer that swims to one point and back at a constant speed. In the chart below and even visualizing the numbers yourself using the above drawing you can see how the effect occurs. Additionally you can see how the effect decreases as the difference between the swimmer speed and the river flow grows. Finally you can see that length does not effect the relationship. The total time travelling (and path length by the swimmer's estimation) grows by a certain percentage based on the prior mentioned relationship.

One Direction path length 10 20
Swimmer speed 2 2
Flow 1 1
Norm (no flow) 10T/20L 20T/40L
Swimmer's Est. Time (*speed = total 2-way length) 13.333 26.666
Relation to Norm 133.333% 133.333%
One Direction path length 10 20
Swimmer speed 3 3
Flow 1 1
Norm (no flow) 6.666T 13.333T
Swimmer's Est. Time (*speed = total 2-way length) 7.5 15
Relation to Norm 112.5% 112.5%
One Direction path length 10 20
Swimmer speed 6 6
Flow 1 1
Norm (no flow) 3.333T 6.666T
Swimmer's Est. Time (*speed = total 2-way length) 3.42857 6.8571
Relation to Norm~ 102.8% 102.8%
 
 
Now that I have established the relationship, I can deal with the the Earth's speed (flow) and the speed of light (swimmer) in units of their relationship.
 
Earth's Orbital speed: 29.78 km/s
Light Speed: 299,792.458 km/s
10,066.906 to 1
 
One Direction path length 100669.06 (x2 = 201,338.12)
Light Speed (swimmer) 10,066.906
Ether wind (Flow) 1
Norm (no flow) 20
Actual travel time (Swimmer's Est.) 20.000000197350382813136543039538
Relation to Norm 1.0000000098675191406568271519769
 
trip1 -1 = 10.000993452551613337140243511116
trip2 +1 = 9.9990067447987694759962995284223
 
What does this all mean?
 
You can use the "relation to norm" to determine how much longer the path length and/or travel time should be in any MMX-like experiment if there was no such thing as entrainment. The above examples show how distance can scale so long as the relation between the swimmer and flow remain the same. 
 
First we need to examine the device used in the the original MMX  to determine what is meant by path length. If you read the dimentions of device it is is a 1.5m square. By looking at the drawing you can tell that it, at least partially, traverses corner to corner 16 times. If it was exactly corner to corner this would be almost 34 meters. We'll assume that Dr. Michelson has some ability with measurements and only meant that the one-way path was 11 meters but that the total optical path is actually 22 meters. This gives us 217.08 nm extra path length along the direction of the Aether wind. ("Relation to Norm" x Path Length)
 
 
Repeatedly through the publication of the MMX he mentions that they are using yellow light. (577-597 nm)However you'll also find it mentioned in the original publication that a single turn of a 100 thread per inch screw moved "nearly 1000 wavelengths" of what he called white light at that point in the paper.  This screw moved the mirror at the halfway point of the optical path so 1 turn would actually shorten the total path length .02 inch instead of the immmediate thought of .01. This would indicate the light to be 508nm at 1000, or a much as 564nm at 900 wavelengths(which I think he would have said directly) So this is slightly vague. Thankfully he goes on to clarify further: 
"Considering the motion of the earth in its orbit only, this displacement should be 2Dx1.0e-8" (220 nm)Next he goes on to say that "The distance D was about 11 meters, or 2e+7 wavelengths of yellow light; hence the displacement to be expected was 0.4 fringe" This tells us with certainty that he believes his light to be 550nm which is actually green. However, the most important consideration that can tell us for certain what the real wavelength of the light being examined in his white light interferometer experiment was the fact that he used sodium light to line up the fringe system! Since the sodium doublet (special double fringes from sodium interferometry)resides at 589.0nm and 589.6nm and he calibrated his interferometer with sodium light, the darkest fringe was set for 589nm and this differs from his belief about the wavelength. (it's also yellow instead of green) This means that his expectation of .4, while close, is not quite right.
 


 
Instead the calculation comes out to: 217.08nm less 1.34nm that the perpendicular path is modified by aberration gives us a 215.74 path length difference. Divide the actual wavelength of 589 that was used to calibrate the fringes and we have an expected fringe shift of .366 We can forgive his slightly incorrect assumption of a 550nm wavelength given that it was 1887....

For those of you following Michelson's mathematical path that would mean that D is 11 meters / 589nm. (18675721.562 waves) resulting in an expectation of a 0.368566 fringe shift without aberration. (29.78^2 / 299,792.458^2) * 37351443.124 = 0.368566.

 
Aberration
 
This effect is so small that I find it almost unnecessary to write about but because Michelson mentions it at the beginning of the paper, I will also. There is actually a small effect perpendicular to the wind called aberration. However, because it is only a second order effect it calculates to only about 1.34nm extra path length. This should be subtracted from the expected path length difference for ultimate accuracy but it can be ignored for the most part.

 
 
The above figure should reveal the easy misconception on the left and the actual effect on the right. The above drawing should suffice for puzzling it out yourself. The important thing to remember is that the concept picture Michelson represents at the beginning is misleading. The total path during which aberration is happening goes both with and against the wind for the round trip on the actual interferometer instead of the idealized concept. The red path on the left in the above drawing is propagating slightly into a headwind and the blue path below it is propagating with a tail wind.
 
 
 
What Was Actually Seen
 
 
A quick substitution of a few numbers reveals the nature of what was actually detected by the interferometer. Let's suppose that the differential speed we expect between the earth and the ether is around one-third what Michelson expected.
 
Earth's Orbital speed: 10.00 km/s
Light Speed: 299,792.458 km/s
29,979.2458 to 1
 
One Direction path length 299,792.458 (x2 = 599,584.916)
Light Speed (swimmer) 29,979.2458
Ether wind (Flow) 1
Norm (no flow) 20
Actual travel time (Swimmer's Est.) 20.000000022253001145832171615753
Relation to Norm 1.0000000011126500572916085807877
 
 
So, 24.4783nm path length difference should have been the expected reading from this assumption.("Relation to Norm" x Path Length) As you can see, this is a little more than one-tenth the expected reading of the MMX while the actual speed is over one-third as large! 24.4783/589 = .04 of a fringe
 
 
Or lets check another similar number...
 
Earth's Orbital speed: 8.00 km/s
Light Speed: 299,792.458 km/s
37,474.0573 to 1
 
One Direction path length 374,740.573 (x2 = 749,481.145)
Light Speed (swimmer) 37,474.0573
Ether wind (Flow) 1
Norm (no flow) 20
Actual travel time (Swimmer's Est.) 20.000000014241920689623184063047
Relation to Norm 1.0000000007120960344811592031524
 
trip1 -1 = 10.000266858396952842169085573917
trip2 +1 = 9.9997331558449678474540984891303
 
So, 15.6661nm path length difference should have been the expected reading from this Scenario. ("Relation to Norm" x Path Length) As you can see, this is still around one-tenth the expected reading of the MMX while the actual speed is still nearly one-third as large. 15.6661/589 = .027 of a fringe...
 
 
If we look at the number from the page which shows the properly temperature adjusted numbers we get the following readings of the screw head (using the full rotational reading for consideration of the full period effect):
 
Day Median: 42.58125
Lowest deviation from mean: 1.80125 (40.78)
Highest Deviation from mean: .1.77875 (44.36)
Avg deviation: 1.79 of a screw head devision
 
Night Median: 50.863125
low: 1.663125 (49.20) 
high: 1.686875 (52.55)
Avg deviation: 1.675
 
Final Avg: 1.7325
 
In consideration that a single number on the measurement device represents an average .02 (.025 - .0166) of a fringe/wavelength, .03465 of a fringe is what was detected. Slightly less than 10 km/s is what the original MMX detected.
 

A Null Result

To actually qualify as a null reading it would have to be below 5% of the expected wind speed, so let's examine that now:

Earth's Orbital speed: 1.489 km/s
Light Speed: 299,792.458 km/s
201,338,118.2 to 1
 
One Direction path length 2,013,381,182.0 (x2 = 4,026,762,364)
Light Speed (swimmer) 201,338,118.2
Ether wind (Flow) 1
Norm (no flow) 20
Actual travel time (Swimmer's Est.) 20.000000000000000493375960986189
Relation to Norm 1.0000000000000000246687980493095
 
trip1 -1 = 10.00000004966769402172595661881
trip2 +1 = 9.9999999503323064716500043673796
 
In this case you have a grand total of 0.00000054271nm extra path length or a 9.214e-10 of a fringe shift. Only readings smaller than this can be considered truly null by typical standards.
 
 
 
 
The Variation of the Readings
 
Another expectation that is not discussed in the relativistic community is the way the readings would change as the interferometer was turned in a full circle if there was an ether wind. This is a very outstanding behavior because it is a half period effect. This means that a full iteration of some cyclical event happens in only one half turn of the device. The wind was expected to cause the readings to both instead peak and trough twice during one full rotation.
As you can see above, as both arms of the interferometer are at some variation of 45 degrees to the wind, both paths are affected equally, bringing the fringes back the the center. The expectation of this behavior is likely the single most important fact to understand when considering the implications of Michelson's and Miller's results. Both of them had this very distinct pattern even after averaging many many readings together. This cannot be mistaken for random error in readings or errors in the device. (But I suppose I've found that faith can always find a way to twist simple fact)