Doppler-Sagnac: A Surrebutal
Mr. Driscoll is right, and I apologize for my statement that "the effect is due entirely to an unequal change in path length." Such a statement implies knowledge which can never be attained. However, the statement was made in the introduction of my letter, not its conclusion, and such loose speech is certainly more tolerable (though not excusable) in an introduction than in a conclusion, where I made no such remark. The result can be explained in this way (as a path length change), but whether such is the case can never be proved, only disproved. Such is the nature of science. However, I can reaffirm that the Sagance effect is entirely unrelated to Doppler, as explained below.
I assume that Driscolls term a is the radius of the Sagnac interferometer. Let us consider the four mirror case, a quadrant of which is illustrated by Fig. 1. (Extrapolation to more mirrors is trivial, involving only a little bit more trigonometry.)
Figure 1. Quadrant of four-mirror Sagnac interferometer.
Note that both mirrors share the same velocity component, v|| parallel to the light propagation. So between mirrors M1 and M2, there is clearly no relative motion in this direction, regardless of the direction of the light path. Relative motion in this direction would be responsible for any first-order radial Doppler effect The two mirrors do have equal and opposite transverse components of motion, so there is a Doppler effect here, proportional to (1-(2v^/c)2)-1/2. It is critically important to note that this effect is not first-order, and furthermore the Doppler shift is to the red and identical for both directions of light travel. So there is no overall, cumulative Doppler effect that could account for the observed first-order Sagnac effect.
One can detect a slight simplification in the above analysis: The figure does not take into account the fact that during the transit time of the light from one mirror to the other, the receiving mirror will have moved slightly with respect to the transmitting mirror. A more detailed analysis would produce more complicated mathematics with some corrections for this motion during transmission. But when one also includes the effects of aberration when comparing the angle of transmission from one mirror to the aberrated line-of-sight at the next, one finds no net first-order corrections. The effects Driscoll discusses, if they existed, would be first-order effects, so the simplification is justified. One finds again (or still) that the Doppler effect plays no role in the Sagnac effect.
 Renshaw, Curtis E., "Fresnel, Fizeau, Michelson-Morley, Michelson-Gale and Sagnac in Aetherless, Galilean Space," Galilean Electrodynamics, Volume 7, 103-108 (1996)
 Renshaw, Curtis E., "The Radiation Continuum Model of Light and the Galilean Invariance of Maxwell's Equations," Galilean Electrodynamics, Volume 7, 13-20 (1996)
Of course, as Dr. Whitney states, all these analyses take place in an inertial frame, not the co-rotating frame of the apparatus. When trying to decide what is an inertial frame, we are back to Machs question if the Sagnac experiment were the only item in the universe, would the fringe shift appear? The Sagnac experiment compares the fringe pattern seen "at rest" with that obtained after placing the apparatus in motion. One could equally well begin with the apparatus "spinning," then compare it with the fringe "shift" that appears when the apparatus is brought "to rest." The point here is that the apparatus itself must undergo a change of state to obtain the desired effect, namely, a "change in fringe pattern."