In order to open the second form using the Application.Run, once the user
successfully logs in, exit from the thread on which login form is running
using
'Application.ExitThread()' and spawn a new thread. The thread will be
spawned using System.Threading.Thread.  Now another form can be run using
Application.Run on newly spawned thread.

Sample Code:
The following code for the button click event of the 'login button'  enables
the user to log in and run the second form using Application.Run as well.


                           System.Threading.Thread td;
        private void button1_Click(object sender, EventArgs e)
        {
           
            if(this.textBox1.Text=="a" && this.textBox2.Text=="a")
             {
               
               
                Application.ExitThread();
                Thread td = new Thread(new ThreadStart(ss));
                td.Start();

            }   

        }
        protected void ss()
        {
            MessageBox.Show("new thread is running");
            Application.Run(new Form2());
        }