Let's say you know all 3 cartesian components of a particle's velocity and all 3 for it's acceleration. You can split the acceleration vector into two vectors, one parallel (longitudinal) to the velocity vector and one perpendicular (transverse) to the velocity vector. Then, I found the x-components of both acceleration vectors in terms of the 6 variables listed at the start. That would be the component of each acceleration vector parallel to the x-axis. I used the dot product to derive it. As you can see below, I multiplied the transverse-x acceleration component by gamma and the longitudinal-x acceleration component by gamma cube. These are the formulas for the longitudinal and transverse masses. I can get a formula for the x-component of the Force. In order to get the second formula, I differentiated the x-component of the 3-momentum with respect to time. I expected both equations to be equivalent, but try as I might, I can't make them equal. Can you help me spot the error(s) in my formulas?