I'm posting a late response to Charles's message above, for the benefit of anyone currently reading this forum:
Charles says "Dead caps means dead motherboard. The thing is, if there are dead caps your system should not start up. Period."
Wrong. Some dead capacitors in some circuits won't prevent a computer from starting up. It depends on which capacitor and which circuit. A dead capacitor can cause various symptoms after startup, like crashing, freezing, video problems, etc., as people are reporting here. If everyone's eMac that has leaking capacitors just wouldn't start up, nobody reporting here whose Mac has leaking capacitors, would be reporting anything but an eMac that wouldn't start up, but instead they're reporting crashes, video problems, etc.
And, a leaking capacitor isn't necessarily automatically dead. It may have some life (capacitance) left in it, just enough to allow the circuit it's in to still function to some degree, until some other factor kicks in, like the Mac heating up after it's been on for a while, or some process, video action, etc. causing the bad capacitors to be "used more", or used in a way, that they're not being used at other times while the Mac is powered up, causing one of these problems to start to show.
As someone else posted below, the leaking capacitors in the eMacs, and in the iMac G5, were apparently from a batch made using electrolyte from a stolen formula that didn't include a stabilizer chemical. Apple's approach to this problem at first seemed to be, if they weren't responsible for the part being manufactured improperly, they couldn't be held liable (hence the infamous Apple support phrase, "Not a known problem"). It's nice that Apple finally decided to replace, at no charge to the Mac's owner, the eMac and iMac logic boards that had these bad capacitors, though the threat of legal action against Apple by some Mac owners was probably what really did the trick.
Power Mac G4 Mirrored Drive Door, dual 1.25 GHz Mac OS X (10.4.9) 2 gig RAM, two serial ATA drives connected to a FirmTek SeriTek/1VE2+2 card