18V + 18V Power Adapter = +/- 18V Power Supply ?

I have two 18V power adapters (computer - same brand) and want to wire them for use. If I wire them this way, will there be any concern with the power supplies melting or ruining the circuits?
 

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Try it and measure to ensure you get what you expect.
Make sure the mains fuse to both supplies is about the right size.

The problem occurs where the "ground" of the supply is connected to the Chassis enclosing the SMPS. That Chassis MUST be connected to PE. In this situation you CANNOT series connect the PSUs.
This applies to the SMPS that power desk top computers.

But where the PSU is double insulated to ClassII standards and has no PE to connect to, then you CAN series connect the PSUs. As many as you like, eg. four 19Vdc laptop supplies to give +-38Vdc to power an amplifier.