node.js and socket.io - broadcast is undefined

Using Node.js and socket.io, the following works fine unti trying to broadcast back to the clients.

I am getting broadcast undefined?

var http = require('http'),
    sys  = require('sys'),
    fs   = require('fs'),
    io   = require('socket.io');

var server = http.createServer(function(request, response) {
  response.writeHead(200, {
    'Content-Type': 'text/html'
  });

  var rs = fs.createReadStream(__dirname + '/index.html');
  sys.pump(rs, response);

});

var socket = io.listen(server);
debugger;
socket.on('connection', function(client) {
  debugger;
  var username;

  client.send('Welcome to this socket.io chat server!');
  client.send('Please input your username: ');

  client.on('message', function(message) {
    if (!username) {
      username = message;
      client.emit('Welcome, ' + username + '!');
      return;
    }
    socket.broadcast.send('a message');
    //io.socket.send('a message');
    //io.socket.emit('message', username + ' sent: ' + message);
  });

});

server.listen(4000);