Firstly you need to clarify whether you are going to retain your original trade name assigned to you. If the other partners are also are practicing in their individual trade name, there is two options available to you:
Option 1: Either of you who hold the seniority of your practice can merge with the others firm. Then in that case, you need to download the merger form i.e., "FORM E" and prepare / type it on a Non-judicial stamp paper of the value which is applicable in your state and send it to the Decentralized office. Later fill form 18 and agreement copy and send it to the decentralized office.
NOTE: Opt for the seniority option only if you think that other partner will be extremely transparent and trustworthy. Else, this could be a stumbling block when you start growing in the profession, chances of discontent and egos may creep in, hence beware.
Option 2: If you are going for a completely new set up, as a partnership firm, then first you need to apply to the Institute in Form 117 for approval of trade name and send it to New - Delhi Office. Once, trade name is approved you can file form 18 and partnership agreement which has to be sent to the Decentralized Office.