How do I connect Arduino directly to MySQL server using the URL?
You are using DHCP, so this step might be optional.
Specify your network addresses (actually we want to specify DNS only but have to specify IP in 2nd argument, so likely have to specify gateway and subnet too):

IPAddress dns_addr( 8, 8, 8, 8); // GOOGLE DNS server, could be any DNS
IPAddress gateway_addr( , , , );   // put your gateway addr
IPAddress ip_addr( , , , );   // put your IP addr
IPAddress subnet( , , , );   // put your subnet mask

void setup(){
                  Ethernet.begin(mac_addr, ip_addr, dns_addr, gateway_addr, subnet);...


Now you could modify Connector::mysql_connect method (or create new method) as follows (only function declaration is changed, body is not):

boolean Connector::mysql_connect(/*IPAddress server*/const char *server, int port, char *user, char *password)
{
  int connected = 0;
  int i = -1;

  // Retry up to MAX_CONNECT_ATTEMPTS times 1 second apart.
  do {
    delay(1000);
    connected = client.connect(server, port);
    i++;
  } while (i &lt; MAX_CONNECT_ATTEMPTS &amp;&amp; !connected);

  if (connected) {
    read_packet();
    parse_handshake_packet();
    send_authentication_packet(user, password);
    read_packet();
    if (check_ok_packet() != 0) {
      parse_error_packet();
      return false;
    }
    print_message(CONNECTED);
    Serial.println(server_version);
    free(server_version); // don't need it anymore
    return true;
  }
  return false;
} 


Modify the header file also:

boolean mysql_connect(/*IPAddress server*/const char *server, int port,
                                 char *user, char *password)


Now you can supply URL:

if (my_conn.mysql_connect("http:\\www.mysqlserver.com", 3306, user, password))...