Also required:
small reel (about 50g) of 0.7mm enameled copper wire
small reel (about 50g) of 0.015mm enameled copper wire
enclosure (plastic box or a piece of 40mm diameter PVC waste pipe)
PP3 battery clip
a few feet of 16/0.2mm flex or similar
small piece of plain veroboard (no copper strips)
two probes (the heads of 6inch nails)

The circuits.
The inverter transforms the 9 Volt DC into about 5kVAC. The multiplier circuit
is comprised of many stages, each of which adds on the input voltage. You only
need about ten stages to produce 50-60kV. This will produce an arc about 1 inch long.

The Inverter.
The transformer (Phillips RM10 kit or similar) has one primary and two
secondaries. The primary (L1) and one of the secondaries (L2) has 7 turns
of .7mm enamelled copper wire. The other secondary (L3) has around 1000
turns of 0.015mm enamelled copper wire.
Wind L3 first, L2 second and L1 last.
Try to make each winding parallel with the previous. Keep the windings neat and don't
stretch the wire too tight.

the transformer windings are;
L1      7 turns of 0.7mm ECW (enamelled copper wire)
L2      7 turns of 0.7mm ECW
L3      approx. 1000 turns of .015mm ECW
(make sure that you leave about
3mm all round for L1 and L2.)

The inverter transformer is made from a Philips RM10 kit (2 ferrite
cores, an eight pin bobbin and a pair of clips).
Wind L3 first using 0.015mm ECW, wind as many turns as you can onto the
bobbin (700-1000) allowing some space (about 2mm all round) for the two
remaining coils. (I use an old hand drill with the chuck removed as a winder
and I just clamp it to a table)

Critical to successful operation:
Spray lots of acrylic lacquer into the L3 coil, allow to dry for a day or
so and repeat. This will prevent the high electric field
causing internal arcing.

Wrap 1 turn of insulating tape around the coil.
Wind the primary (L1) using 7 turns of 0.7mm ECW, and the other
secondary (L2) also using 7 turns of 0.7mm ECW.

Make sure that there is a small spark gap (about 1-2mm) across the output
of the high voltage coil (L3). This will reduce the electric field strength within
the coil and give it a longer working life.

The resistor values in the inverter are approximate only. You WILL need
to tune the finished circuits to their resonant frequency by varying R2.
You may also be able to reduce the circuit current by varying R1.
When the inverter is connected to the multiplier, arcs may be present
across the spark gap on the inverter output. This is normal.
If the inverter doesn't oscillate, reverse the connections to L1.

The Multiplier.
Build the multiplier circuit on the plain veroboard ensuring that the components
are well spaced out. (at least 3mm between each stage)
When the multiplier has been built, it can be tested by
immersing in vegetable oil (to prevent internal arcing)
and connecting to the inverter.
If the circuits works ok, clean thoroughly with detergent
and dry completely before potting.
Assemble the multiplier into the box after attaching the probes and the wires that
will connect to the inverter. Prepare the potting compound and pour into the box
ensuring that the inserter is fully submerged. Tap the box to dislodge any bubbles.
(the multiplier circuit must be potted in the epoxy to
prevent internal arcing.)

The circuit produces a one inch arc at around 10 cycles per second.
The supply current should only be 250mA from an 8.4V-9V supply.