Argument #1
The Court should rule in favor of Plessy.  Plessy has an equal right to ride in the "white" car since he was sold and purchased a ticket for that car.

Argument #2
The Court should rule in favor of Ferguson in this case. There existed a state law which said that the railroad must set up separate but equal facilities for the white and colored races. The railroad has done this, therefore Plessy must sit in the "colored" car.