I did not intend this posting for didactic analysis; but rather was putting it as a hyperquick posting (in terms of my time) and intended it to be merely a quick informational posting. This means I did not bother to define terms; thus leaving potential for confusion.
The author makes clear that u is the usual 3velocity in some frame (actually one component of the 3velocity where the spatial axis is aligned with the direction of the 3velocity; meaning we are effectively dealing with a spatially one dimensional situation and a 1+1 dimensional spacetime). The equation is therefore writing a relationship for one component (a spatial component) of the 4momentum. IMO, it is a scalar equation and not susceptible to criticism on dimensional grounds. (see the textbook for other assumptions)
I agree with JD, that my posting gives the appearance of such confusion, since I didn't write a careful posting clearly defining the symbols.
_________________________
 Original Message
 From: physlbounces@carnot.physics.buffalo.edu [mailto:physl
 bounces@carnot.physics.buffalo.edu] On Behalf Of John Denker
 Sent: Friday, January 27, 2012 4:57 PM
 To: Forum for Physics Educators
 Subject: Re: [Physl] Relativistic Mass or lack thereof in intro books

 On 01/27/2012 01:13 PM, Rauber, Joel wrote:
 > I just received a copy of the 3rd edition of Randall Knight's Physics
 > for Scientists and Engineers  with modern physics
 >
 > And was pleased to see that he eschews relativistic mass.

 Very nice.

 > i.e. his momentum for a particle is
 >
 > P = m *(Delta x over Delta Tau) =gamma*m*u

 Hmmmmm. Let's take apart the double equation on the last line:

 P = m *(Delta x over Delta Tau) [2]

 That is entirely reasonable and conventional. It is an equation
 involving the 4vector momentum P and the 4vector position x.
 So far, so good.

 Meanwhile, I am confused by this part:

 m *(Delta x over Delta Tau) =gamma*m*u [3]

 Conventionally,
 v = dx_S/t = classical 3vector velocity (in some frame)
 where x_S is the projection of the position vector (x)
 onto the spatial part of the given frame,
 and the denominator involves t, the projection onto
 the timeaxis of the given frame.
 u = dx/dτ = the 4vector velocity (in all frames)
 where the denominator involves τ i.e. the proper time

 If we take the u in equation [3] to represent the 4velocity, the
 factor of gamma is wrong there. The usual 4vector equation is
 simply
 P = m u [4]

 If we take the u in equation [3] as a typo or as an unconventional
 representation for the classical velocity dx/dt, the equation doesn't
 make sense, on dimensional grounds. That is: it appears to have a
 4vector on the LHS and a 3vector on the RHS. Similarly, the LHS
 is valid in all frames, while the RHS only makes sense in some chosen
 frame.

 For more on how to handle this topic correctly, see
 http://www.av8n.com/physics/spacetimewelcome.htm#secvelocities
 momenta
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