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To summarize,
When force and velocity are perpendicular, F = gamma m a.
When force and velocity are parallel, F = gamma^3 m a.
The product (gamma^3 m) was at one time referred to as the "longitudinal
mass", as distinguished from (gamma m), the "transverse mass". To me
the non-existence of one simple "relativistic mass" covering all cases
is a good reason to join (or stay in) the "anti" camp.