It is stated that the Lord Vishnu has taken Nine incarnations so far for different purposes and His 10th incarnation "Kalki" is yet to take place. In the nine Avataar some authors include Lord Balaraam while of lately some are including Buddha in place of Balaraam. Will some learned persons clarify the correct position? I am not well versed in Puraan but to my mind either of the inclusions do not appear to be logically correct as of Vishnu's ninth Avataar for the following reasons:

Balaraam is said to be the incarnation of Aadi Shesh (Lord Vishnu's snake companion) and, therefore, cannot be legitimately claimed as an avatar of Lord Vishnu. Further, in the Puraan it is mentioned that the nine Avatars took place at different periods except the avatar of Lord Parashuraam who gave way to Lord Raam in quick succession. Therefore, tt is not understood why there were two Avataar of the Lord as Krishn and Balaraam in the same period that too as brothers. Further, if Lord Balaraam, an incarnation of Aadi Shesh, can be considered as one of the nine Avataar, why Lakhman (brother of Lord Raam), said to be an incarnation of Aadi Shesh, is not considered as an Avatar?

As already stated above, I find the inclusion of Gautam Buddha as the 9th avatar, in place of Balaraam is a recent phenomenon and a matter of convenience. Lord Buddha's birth was about 2500 years back, very much part of this Kali Yug period whereas in the Puraan it is stated that only Kalki Avataar will be borne as the 10th Avataar in the Kali Yug. There was only one time in Tretaa yug when Vishnu's two Avataar were living together. And as soon as both met one had to leave. Parashuraam Jee was there before Raam, and Raam's purpose was not yet completed, so Parashuraam Jee had to leave.

Why Buddha is Not an Avataar of Vishnu
With the basic principle, there were 5 Avataar who came for a short while just to complete a specific task. That is why they could be called task-oriented incarnations -
(1) Varaah (Wild Boar) Avataar to bring Prithvi up from the water, on the way he killed Hiranyaaksh and went away
(2) Matsya (Fish) Avataar to keep Manu alive in the flood and went away
(3) Kashchap (Tortoise) Avataar to support Mandaraachal Parvat for Saagar Manthan and went away
(4) Narasinh (Man-Lion) Avataar to kill Hiranyakashyap and went away
(5) Vaaman (Dwarf) Avataar to take Swarg back from Daitya Raaj Bali to give it to Indra and went away

There were 3 Avataar, (6) Parashuraam, (7) Raam and (8) Krishn, who also came with a purpose, but still lived on Prithvi for quite some time. They did not take birth, they just appeared. Regarding their disappearance, from among above Avataar all Avataar disappeared then and there, and among these three Avataar, Raam disappeared in Sarayoo River, and Krishn went to His Dhaam along with His mortal body (Bhaagvat Puraan, 11/31), but Parashuraam Jee, as it is believed, is still living and will train Kalki Avataar in Martial Arts. It shows that no Avataar took birth and no Avataar was cremated or buried. While Buddha was supposed to be born and died and was cremated.

Further, though Hindu believe that Buddhism is an offshoot of Hindu religion, the followers of Buddha do not agree with that theory. Again, though the teachings of Lord Buddha are in consonance with the tenets of Hindu religion, it does not appear logical to include Buddha as the 9th avatar as all the previous avatars starting from Narasinh Avatar to Krishn Avataar were essentially for the purpose of eradicating evil persons and protection of Dharm. The coming avatar Kalki is also supposed to be for the same purpose and, but Lord Buddha, with his wonderful teachings of Ahinsaa and renunciation etc appear to be on different plane altogether. We must remember that Lord Mahaaveer too espoused similar principles and is worshipped by Jains.

Then who is the 9th Avataar?