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I could give you directions of hooking the two switches together to make a 'mechanical AND' switch... but just what exactly is supposed to be let through, or given to the applicator to make it cycle... ? ... perhaps +24V ? ... perhaps something else?
Let me know also, whether when one switch is pressed, the normally open is connected to the normally closed, or whether the normally open is connected to the +24V, or what?
There will be two switches.
Switch # ***** will be wired to + on the Brown wire.
and to GND on the Blue wire.
Switch # ***** will be wired from the Brown wire (which must NOT be connected to anything else but this, not to + for sure) ... from the Brown wire on switch # ***** back to the Black wire on switch # *****
Switch #2 will otherwise be wired from the Black wire on it, to the pin # ***** on the applicator.
Here's a primitive sketch to make all clear as possible: