Within a few hours a user had suggested it could well be Coenosia tigrina, one of a group of killer flies. Putting the name in Google and looking at the photos it shows I am very sure this is the fly I saw and it certainly looked as though it was attacking the other fly. Some information about the killer fly can be found on BugGuide.
The iSpotNature site looks to be an excellent resource for finding the identity of many unknowns in the natural world. What is the iSpot site?.... This is from their help page:
"iSpot is a website aimed at helping anyone identify anything in nature. People upload their observations of wildlife, help each other identify it, and share and discuss what they've seen.
A good place to start is to look at some of the latest observations - you can see what people are observing right now. Anyone can see what's on the site without registering - just go back to the home page, and click on the photos for the latest observations.
We'd love you to make use of the site yourself, by showing your own observations, or commenting on other people's and helping them to identify what they've found. To do this you'll need to register on the site - this is totally free, and does not commit you to anything. All you need to provide is a user name, an email address, and a password."