Prostheses are an element of great importance and necessity for many people who have suffered amputations or lack a member of their body; however, prostheses tend to have very high prices, so they are not accessible to all people who want to use them. For this reason, underactuated fingers in prostheses allow a functional and useful alternative for them. In this article, the design and mathematical analysis of an underactuated finger for hand prostheses is carried out. Thus, the finger can be actuated by means of a single motor, being the protheses fingers capable of recovering a pre-established position. The mathematical analysis of the underactuated fingers for prosthesis hands allows checking the movement due to the results of trajectory, velocity, and acceleration. Using the Jacobian Matrix and Newton-Raphson Method proportionate a linear model that helps the subsequent control of velocities and accelerations of each link of the finger. For the model, dimensions of a man and woman hand have been considered. Results are appropriate for hand prostheses, with sufficient velocity but not too high, which helps to prolong the useful life of the mechanism.