If a private company has only one continuing director, he can act by virtue of Section 174(2) of the Companies Act, 2013 to increase the number of directors to meet the quorum. Once the quorum is met, the Board of Directors can appoint additional directors to the company.

The process for approving the appointment of additional directors in such a situation is as follows:

1. Convene a Board Meeting: The continuing director should convene a board meeting by issuing notice to all the existing directors of the company. The notice should include the agenda of the meeting, which should include the proposal for appointing additional directors.

2. Pass a Resolution: At the board meeting, a resolution should be passed to increase the number of directors to meet the quorum requirement, and also to appoint additional directors. The resolution should be passed by a majority of the directors present at the meeting.

3. File Form DIR-12: The company must file a Form DIR-12 with the Registrar of Companies within 30 days of the appointment of the additional director. The form should include details of the new director, including his name, address, DIN, and consent to act as director.

4. Update the Register of Directors: The company should update its Register of Directors and Key Managerial Personnel with the details of the new director.

5. Intimation to Stock Exchange: If the company is listed on a stock exchange, it must intimate the stock exchange about the appointment of the new director within 24 hours of the appointment.

Once these steps are completed, the newly appointed director can start acting as a director of the company. It is important to note that the Articles of Association of the company should also be checked to ensure that the appointment of additional directors is in accordance with the provisions mentioned therein.