The short answer to this question is that when we fly along with a constant Gyro Heading set, but no Astronomic Precession, the rate at which the Transport Wander changes the direction of flight, is exactly the same as the rate at which a great Circle track would change. So we are flying along a Great circle track.
The rather longer explanation is provided below.
A google search for “Astronomic Precession” reveals an explanation which concerns how the axis of rotation of the Earth describes a circular motion over a period of approximately 26000 years. This is obviously not the interpretation intended by the author of the question.
In this question Astronomic Precession is intended to mean Earth Rate Gyro Wander. It is caused by the facts that the Earth rotates about its spin axis and the Meridians converge towards the Poles. The equation for Earth Rate Wander (ERW) is:
ERW = 15 x Sin Latitude in degrees x time in hours
The number 15 is in this equation because the Earth rotates through 15 degrees of longitude during each hour. So for any given time period the equation can be restated as:
ERW = Change in longitude x Sin Latitude.
The condition of “with no astronomic precession” can be achieved for the purposes of this question by assuming that the Earth has stopped rotating. Or perhaps more realistically by designing a Latitude Nut system which automatically adjusted itself for changes in latitude. Either of these solutions would eliminate Astronomical Precession (ER) but would leave Transport Wander (TW) unchanged. The equation for TW is:
TW = East-West ground speed x time of flight x Sin Latitude.
East-West ground speed x time of flight = Change of longitude, so the equation can be rewritten as:
TW = Change in longitude x Sin Latitude…………………..Equation 1.
Great Circles have the following properties:
1. All Great Circle form straight lines on the surface of the Earth and have their
centres at the centre of the Earth.
2. All Great Circles which run in a true North-South direction cross the Parallels of
Latitude at a constant angle of 90 degrees.
3. The Equator is the only Great Circle which runs in a true East-West direction, and
this crosses all of the Meridians at a constant angle of 90 degrees.
4. All other Great Circles cross the Meridians at angles which gradually change as we
move around the circle.
So why is that Great Circles running other than north-south or east-west do not cross the Meridians at a constant angle? The reason of course is that the Meridians are parallel to each other only at the Equator, then converge as latitudes increase towards the Poles. So any straight line (Great Circle) running in a direction other than due north-south or due east-west, must cross successive Meridians at different angles. We can see this if we look at the equations for convergence of the Meridians and the Conversion Angle which defines the direction of the Great Circles.
Convergency = Change in longitude x Sin Latitude
And
Conversion angle = ½ Convergency = ½ Change in longitude x Sin Latitude
The change in direction of a Great Circle track between two points is twice the conversion angle, so:
Great Circle direction change = Change in longitude x Sin Latitude……Equation 2
So we now have:
Equation 1…….Transport Wander = Change in Longitude x Sin Latitude.
Equation 2…….Change in great circle track = Change in Longitude x Sin Latitude
This means that the rate of change of the Great Circle Track between two points is equal to the rate of Transport Wander. So as we fly along with a constant Gyro Heading set, the rate at which the Transport Wander changes the direction of flight, is exactly the same at which a great Circle track would change. So we are flying along a Great circle track.