If this is, as depicted, a series lamp connecvtion RS, the winding arrangement would be as follows:
The primary, conected between the phase input and Neutral, will be in one side spool (likely the left one in this picture).
There will be then three ~3.5V filament winding on the same spool with the primary, one (the blue wires) connected to the Neutral (very likely being directly the bottom section of the primary winding, so one of the ble wires would be directly connected to the Neutral wire). Second filament winding will be just isolated (for the black lamp wires), the last filament winding (for the reds) would be connected to the end of the secondary.
The secondary winding would be either just on the second (right hand side) spool (if the ballast is designed as 230V), or have part of the winding on the spool with primary, the rest on the right hand side spool (for a 120V fed ballast), in any case it is connected as a continuation of the primary, from the phase terminal.
So the filaments are powered with low impedance 3.5V, the arc are fed from the whole main winding, where half of it is on the left and half on the right spool. The magnetic shunt between the two spools forms a high series equivalent reactance of the secondary towards the primary and so forms the main ballasting impedance for the lamps.
The series connection is advantageous in one aspect: The working arc voltage of both lamps sums up together, but the voltage required for their ignition is way less than sum of ignition voltages of both lamps (actually it is about the sum of the top lamp arc voltage and the bottom lamp ignition voltage), what means it is way lower ratio than for just a single lamp.
So for a single F40T12, where you need to count with about 300Vpeak for ignition and about 105V arc voltage, you need a 220V OCV secondary then loaded by 0.43A, so about a 95VA winding for a single F40.
For two F40's operated independently you would need 2x96VA so 192VA ballast size (this figure is, what determines the final ballast size)
But when operated in series the lamps ignite in sequence: First, when there is no current in the arc circuit, one lamp is effectively shorted by either capaitances to ground, or a bleeder (either high value resistor or a small capacitor parallel to one lamp), so all voltage is across the remaining lamp, igniting it. Once that lamp is ignited, the voltage drop across it is about its normal arc voltage (assume the electrodes are already hot, as it is a RS ballast), so all the rest becomes available across the second lamp for its ignition.
So you suffice with 300+105=405V peak voltage, what means about 280Vrms, so with 0.43A load it means just about 110VA size ballast, capable of operating the same pair of F40's. So nearly half of the secondary rating compare to the independent lamp circuit ballast.