What is the belief in the transmigration of the soul?

Some people believe in the transmigration of souls. Do we take some other form and so go on in a constant round of progression? Does the soul enter other bodies sometimes better and sometimes worse than its own?

In one sense, it does enter other bodies and acts through other bodies than the human, and in another sense, it does not. 

The soul expresses itself, I believe, through all that is beneath it. Everything becomes a medium for the soul. 

The granite rocks are the soul's mediums, the skies, the air, the water, the flowers, the beasts, the birds, the fishes—everything that is beneath the soul becomes the soul's agent, and so far as the soul is able to find expression through these forms or by the agency of these forms, so far it becomes incorporated in the form allied to it. 

The ancients grappled with a very great truth in their theory of the transmigration of souls. 

They intuitively perceived the power of the soul over all matter, and perceiving its power, they were led to conclude that it would use the power and, therefore, become incarnated in other forms than the human.

—Professor John Hubbard in Spirit