Yes, you now have it wired for delta.
The supply is connected to ANY 2 terminals out of L1, L2, L3 - but lets say you use L1 and L2.
The capacitor connects between one of those (choice determines direction of rotation) and L3.
Without the capacitor, the winding between L1 and L2 creates a strong field, and the other two windings create a weaker field that is in phase. Inside the stator, that means a field that simply flips back and forth with no rotational component. This is just the same as a single phase motor without a start winding connected. If you were to spin the motor up and then apply power, it would run (just not very well) - this is something I've done in the past (very carefully
) to test that a problem was just the start circuit.
The capacitor creates an out of phase current in two of the windings (those connected to the L3 terminal) and so creates a rotational element to the field inside the motor.
Selection of capacitor. To start with, it must be a run capacitor - continuously rated and not less than 240V
AC. Many capacitors are DC and will simply explode if connected to AC - and trust me, an exploding electrolytic capacitor is an "interesting sight" even for a tiny one (I'd not like to be near a big one
). It's a matter of trying different sizes until you get the currents through the windings as balanced as you can. I don't know if there are any rules of thumb - I've only done it by trial and error (more error
)
Adding some capacitance between L1 and L2 will probably reduce supply current by compensating for the reactive power factor drawn by the motor - but don't worry about that.
Had to go back and check as I'm following more than one thread involving motors. As you are starting under no load, you may be able to get away without any special starting arrangements - you are only starting up the inertia of the motor without trying to start any load at the same time. If you were starting under load then you'd need extra capacitance during starting.
This video will give you some idea about what you are building - except that what this guy calls the pony motor is your sewing machine motor. Just be aware that some of his working practices are "poor" and of questionable safety
What he refers to as a "potential relay" is what would also be known as a voltage sensitive relay.