Several months ago my two beagles dug into our neighbors yard in the middle of the night and were attacked by two rottweilers. We were able to get them out but the smaller beagle had to be put down. The other beagle seemed depressed at first, but she showed improvement after a few weeks, especially after we got a new beagle puppy. At first she warmed up to the new puppy and played with her several times a day. But there have been random times where this older dog attacks the new one viciously. We have tried ignoring the puppy and giving her lots of attention, giving her extra treats, and allowing her to have privileges that she had not had before, but the attacks are still happening. We have tried to pinpoint causes---petting times, treat times, etc., but cannot seem to find a pattern because sometimes there are attacks when there seems to be no stimulus. How can we better this situation?

I'm so sorry to hear about your beagle. Losing a dog is painful, especially like this.

Unfortunately, your older beagle has become dog-aggressive.
It's an unfortunate, but common reaction to having survived being attacked by other dogs.

She isn't jealous of the new puppy- she's afraid. It's an "attack first, or be attacked" mentality.

There really isn't any one cure-all for the problem. Some dogs can be helped, some can't.

I would suggest reading Ed's articles about Dog Aggressive Dogs for some info:

http://leerburg.com/dog-agg.htm
http://leerburg.com/qadogfight.htm
http://leerburg.com/introducingdogs.htm


Your best bet for now, and possibly for good, is to keep the dogs separate. If the older dog isn't used to a crate, it may be easier to crate train the new puppy, and keep the puppy in the crate while the older dog is around.

It's very important not to let your older dog attack your younger pup, or you'll end up with two dogs, with the same exact problem.

I would suggest an immediate search for a local trainer EXPERIENCED in aggression.
Crates x 2 in the meantime while you search for a trainer(not one for only one dog, or you will end up with all kinds of other issues.) The older dog needs structure to understand aggression is NOT tolerated at all - a crate will help with that. If it's not used to a crate, it's time to get the dog used to it. I would recommend NO loose dogs at all - tethered to you on a leash for BOTH of them at separate times. One in crate, one out, switch as needed.
Don't allow the dogs contact with one another at ALL. Even if they seem "better" until you have someone experienced there.
Trying to "fix" the problem by ignoring the pup is going to cause issues with that pup, first off, and it is strengthening the older dogs' dominance. extra privledges, bribes (treats) etc are also strengthening the problem and making it worse.
Bottom line - it doesn't MATTER if the dogs like one another. It's not up for debate. They have to learn to respect YOU and your leadership. You decide to bring in a dog into the home, it's your decision! It's their job to adapt. If they are unable to do that, well then it's time to make changes permanately in the home so that they are not together and hurting one another.