Inspiron 2350 Power cycling issues

Inspiron 2350 all in one desktop-  When powering up the unit will log in and start to go through normal operations.  Then all the sudden it will shut down, no log out process or warning, just immediately shuts off.  When you try to restart the unit by pressing the power button it will start the process again then shut off immediately again.  Repeating the cycle the power up time reduced each attempt till it will not react at all.  I can unplug the power supply for an hour or so and it will repeat the above.  Leaving the power supply unplugged for longer periods days at times will allow the unit to power up and get to the desktop screen after a random period of time the unit will shut down again.

I have tested the power supply with a multi meter and it seems to be operating normally.  the unit is only approx 18mths old.

Any suggestions as to the problem and or solution?