I'd like you to check some components on the motherboard.
Specifically, Electrolytic Capacitors.
More specifically, the one's used in the motherboard Voltage Regulator Circuit.
This is a leading cause of a computer booting up, then shutting down, then restarting again.
eMachines are a budget computer. The low cost, is due to low quality hardware components inside the computer..
Electrolytic capacitors on the motherboard, being some of the above stated components.
1) Electrolytic Capacitors slowly build a charge up, then release the charge all at once.
Akin to a swimming pool slowly being filled up by a garden hose, then one wall of the pool is taken down all at once.
The Electrolytic capacitors used on the motherboard, are aluminum Electrolytic Capacitors, and are Radial in design.http://en.wikipedia.org/wiki/Electrolytic_capacitor
Viewing the second photo down, at the top right of the page, the bottom Electrolytic Capacitor is of the radial design.
Both leads come out of the bottom.
Essentially, an aluminum Electrolytic Capacitor is a small aluminum 'can', with three strips of thin foil rolled up together, inside.
A) One strip is metal, and is the Conducting strip.
B) One strip is also metal, but has a non-conducting medium applied to it.
C) The last strip is composed of a paper-like material, and is soaked with Electrolytic Paste.
The paper strip is placed between the two metal strips, and all three strips are rolled up tightly, then inserted into the 'can'.
(Of the two leads that protrude from the bottom of the capacitor, one is a Positive lead, the other is a Negative lead.
The Positive lead is connected to the Conducting strip.
The Negative lead is connected to the Non-Conducting strip)
At the bottom of the 'can' case is a seal. This seal is composed of a synthetic rubber-like material, and is a flat disk shape.
At the top of the 'can' case is another seal. It is a thin, flat, disk of metal.
The center of this disk shape has an X, or K, etched partially into the disk.
Electrolytic Capacitors break down over time.
The design manufacturer of a product that uses this type of capacitor, is aware of this, and plans accordingly.
The capacitor used is 50 percent better than is required.
This way as the capacitor breaks down, it weakens to a state that is still usable.
Low quality Electrolytic Capacitors have inferior Electrolytic paste.
As the paste breaks down a gas is formed. (Hydrogen gas)
The gas expands inside the can's case, and slowly pushes the Electrolytic paste out.
When the capacitor is starting to break down, the outside can case bulges.
As the capacitor breaks down further, the paste is pushed out of the bottom seal, (Rubber like disk has one side pushed out of the bottom), and/or breaks the etched design on top open, and paste pushes out.
So much paste loss, and the capacitor can operate in a weakened state.
Too much paste loss, and the capacitor fails.
The paste can also dry up inside, and will show no outward visual signs of failure.
Computer unplugged from power, computer case open.
TOUCH (Not 'shouting') the metal frame of the open computer case, BEFORE you reach inside, to relieve your body of static electricity.
[Static WILL fry out (Short Circuit), the delicate hardware components inside a computer.
You may not even see it, or feel it.
Computer unplugged from power you are safe.
TOUCH the metal frame, and your computer is safe.
Work on a table.
Do Not work on a bed, or directly on a carpet floor. These are high areas of static.
Should you get up, and walk away in the middle of working on your computer, be Sure to touch the metal frame upon your return]
See if you can observe visual signs of capacitor failure.http://www.capacitorlab.com/visible-failures/index.htm
A video showing the results of capacitor failure, in the motherboard Voltage Regulator Circuit,http://www.youtube.com/watch?v=KN6_-8fYHo0
(Starts at 0:04)
Part of what the motherboard Voltage Regulator Circuit does, is to regulate voltage for the Processor.
The Processor must have a steady, clean, supply of voltage, and the voltage must be within a certain range.
Too much, or too little voltage, and the Processor turns off.
(BIOS turns the Processor off)
Failing Electrolytic Capacitors in their weakened state will charge up, and release for a brief time period, but then will break down.
This is why the computer comes on, then shuts down.
Replacing the motherboard is the solution.
One more 'ailment' may apply.
It could be Electrolytic Capacitors in the Power Supply, that are breaking down.
Test the Power supply voltages, or replace the Power Supply with a KNOWN to be good, working unit that is compatible, for a test.
(Compatible being the Power Supply is the correct size, and shape, plus also has enough Wattage, and the correct amount of power cables, plus the correct power cables)
To test you will need a multimeter, or a power supply tester.
A computer (Personal computer) Power supply puts out three main voltages.
A) 3.3 Volts. (Wires with Orange insulation)
B) 5 Volts (Wires with Red insulation)
C) 12 Volts (Wires with Yellow insulation)
If the 12 Volt power rail is 11 to 13 Volts, the power supply is okay.
Below 11 Volts requires replacing the Power Supply.
You can post additional questions in a Comment.